Involutions of Azumaya algebras
Abstract.
We consider the general circumstance of an Azumaya algebra of degree over a locally ringed topos where the latter carries a (possibly trivial) involution, denoted . This generalizes the usual notion of involutions of Azumaya algebras over schemes with involution, which in turn generalizes the notion of involutions of central simple algebras. We provide a criterion to determine whether two Azumaya algebras with involutions extending are locally isomorphic, describe the equivalence classes obtained by this relation, and settle the question of when an Azumaya algebra is Brauer equivalent to an algebra carrying an involution extending , by giving a cohomological condition. We remark that these results are novel even in the case of schemes, since we allow ramified, non-trivial involutions of the base object. We observe that, if the cohomological condition is satisfied, then is Brauer equivalent to an Azumaya algebra of degree carrying an involution. By comparison with the case of topological spaces, we show that the integer is minimal, even in the case of a nonsingular affine variety with a fixed-point free involution. As an incidental step, we show that if is a commutative ring with involution for which the fixed ring is local, then either is local or is a quadratic étale extension of rings.
2010 Mathematics Subject Classification:
Primary: 16H05, 14F22, 11E39, 55P911. Introduction
1.1. Motivation
Let be a central simple algebra over a field and let be an involution, i.e., an anti-automorphism satisfying for all . Recall that can be of the first kind or of the second kind, depending on whether restricts to the identity on the centre or not. We further say that is a -involution where .
Central simple algebras and their involutions play a major role in the theory of classical algebraic groups, and also in Galois cohomology. For example, letting denote the fixed field of , it is well-known that the absolutely simple adjoint classical algebraic groups over are all given as the neutral connected component of projective unitary groups of algebras with involution as above, where varies (here we also allow with the switch involution), see [knus_book_1998-1, §26]. In fact, all simple algebraic groups of types , , , , excluding , can be described by means of central simple algebras with involution. Involutions of central simple algebras also arise naturally in representation theory, either since group algebras admit a canonical involution, or in the context of orthogonal, unitary, or symplectic representations, see, for instance, [riehm_orthogonal_rep_2001].
Azumaya algebras are generalizations of central simple algebras in which the base field is replaced with a ring, or more generally, a scheme. As with central simple algebras, Azumaya algebras and their involutions are important in the study of classical reductive group schemes, as well as in étale cohomology and in the representation theory of finite groups over rings; see [knus_quadratic_1991].
Suppose that is a quadratic Galois extension of fields and let denote the non-trivial -automorphism of . A theorem of Albert, Riehm and Scharlau, [knus_book_1998-1, Thm. 3.1(2)], asserts that a central simple -algebra admits a -involution if and only if , the Brauer class of , lies in the kernel of the corestriction map . Saltman [saltman_azumaya_1978, Thm. 3.1b] later showed that if is replaced with a quadratic Galois extension of rings , then the class lies in the kernel of if and only if some representative admits a -involution. Here, in distinction to the case of fields, an arbitrary representative may not posses an involution. However, a later proof by Knus, Parimala and Srinivas [knus_azumaya_1990, Th. 4.2], which applies to Azumaya algebras over schemes, implies that one can take such that .
The aforementioned results all have counterparts for involutions of the first kind in which the condition is replaced by .
In this article, we generalize this theory to more general sites and more general involutions. We have two purposes in doing so. The first, our initial motivation, is to demonstrate that the upper bound in Saltman’s theorem, guaranteed by [knus_azumaya_1990, Th. 4.2], cannot be improved in general for involutions of the second kind. The statement in the case of involutions of the first kind was established in [asher_auel_azumaya_2017]. Our general approach here is similar to [asher_auel_azumaya_2017], [antieau_unramified_2014] and related works. That is, the desired example is constructed by approximating a suitable classifying space, and topological obstruction theory is used to show that it has the required properties. In contrast with [asher_auel_azumaya_2017] and [antieau_unramified_2014], the obstruction is obtained by means of equivariant homotopy theory.
We therefore introduce and study involutions of the second kind of Azumaya algebras on topological spaces. In fact, we develop the necessary foundations in the generality of connected locally ringed topoi with involution, and show that Saltman’s theorem holds in this setting. In doing so, we stumbled into our second purpose, which we now explain.
Any involution of a field is either trivial or comes from a quadratic Galois extension, which is why the classical theory sees a dichotomy into involutions of the first or second kind. For a ring, the analogous involutions are the trivial involutions or those arising as the non-trivial automorphism of a quadratic étale extension . Geometrically, these correspond to extreme cases where one has either a trivial action of the cyclic group on a scheme, or where the action is scheme-theoretically free. One may also view the free case as corresponding to an unramified map . This dichotomy has been preserved in the literature on involutions of Azumaya algebras over schemes, say for instance [knus_azumaya_1990] and [knus_quadratic_1991], by considering only trivial or unramified involutions of the base ring.
There are, of course, involutions which are neither trivial nor wholly unramified. For instance, one may encounter involutions of varieties that are generically free but fix a nonempty closed subscheme. Alternatively, there are involutions of nonreduced rings that restrict to trivial involutions of the reduction—these are geometrically ramified everywhere, but nonetheless non-trivial.
Our second purpose therefore became developing the theory of -involutions on Azumaya algebras with minimal assumptions on . We establish a generalization of Saltman’s theorem, and present a classification of -involutions into types, generalizing the classification of involutions of central simple algebras as orthogonal, symplectic—both of the first kind—or unitary—of the second.
In more detail, given a field of characteristic not and an involution , recall that two degree- central simple -algebras with involutions extending are of the same type if they become isomorphic after base change to a separable closure of the fixed field of . This definition extends naturally to the case of a general connected ring in which is a unit by replacing “a separable closure” by an étale extension of , the fixed ring of . It is natural to ask how many types are obtained in this manner, and how to distinguish them effectively. In the classically-considered cases of trivial or unramified involutions on , the situation is known to be similar to case of fields: When , there are at most two types — the orthogonal, which occurs for all , and the symplectic, which occurs only for even . When is quadratic étale, only one type, called the unitary type, occurs for all .
We describe the types for arbitrary and give a cohomological criterion to determine when two involutions are of the same type. This criterion implies in particular that the type of an Azumaya algebra with involution is determined entirely by the restriction of to the ramification locus of . More than two types may occur. With this new subtlety, one can further ask, in the context of Saltman’s theorem, what are the types of -involutions which can be exhibited on representatives of a given Brauer class in . Our generalization of Saltman’s theorem answers this question.
To demonstrate some of the ideas above, let us consider a field of characteristic different from and the ring of Laurent polynomials with the involution . The fixed ring is . The map is ramified at two points, and , and unramified elsewhere. Our results show that there are types of -involution for even-degree algebras and type in odd degrees. Furthermore, the type of a -involution is determined by the types — orthogonal or symplectic — obtained by specializing to and . For example, consider the -involution of given by
| (1) |
Evaluating at , the involution of (1) becomes orthogonal, whereas evaluating at makes it symplectic. Our generalization of Saltman’s theorem implies that if is represented by an Azumaya -algebra admitting a -involution, then each of the types of -involutions is the type of a -involution of some representative of .
It seems likely that our results on types could be used to extend the theory of involutive Brauer groups, intiated in [parimala_92] (see also [verschoren_98]), to schemes carrying ramified involutions. We hope to address this in subsequent work.
We finally note that from the point of view of group schemes, the study of -involutions of Azumaya algebras in the case where is neither trivial nor unramified amounts to studying certain group schemes over which are generically reductive but degenerate on a divisor. Specifically, the projective unitary group of an Azumaya algebra with a -involution is generically of type and degenerates to types , or on the connected components of the branch locus of . The study of degenerations of reductive groups have proved useful in many instances. Recent examples include [auel_parimala_suresh_2015] and [bayer_17], but this manifests even more in the works of Bruhat and Tits on reductive groups over henselian discretely valued fields [bruhat_I_72], [bruhat_II_84], [bruhat_III_87]. Broadly speaking, degenerations of reductive groups are encountered naturally when one attempts to extend a group scheme defined on a generic point of an integral scheme to the entire scheme, a process which is often considered in number theory.
1.2. Outline
Following is a detailed account of the contents of this paper, mostly in the order of presentation. While the majority of this work applies to schemes without assuming is invertible, we make this assumption here in order to avoid certain technicalities.
Section 2 is devoted to technical preliminaries, largely to do with non-abelian cohomology in the context of Gorthendieck topoi.
Let be a scheme and let be an involution. Our first concern is to specify an appropriate quotient of by the group . There is an evident choice when with a ring, since one can take the quotient to be , where is the fixed ring of . However, at the level of generality that we consider, there is often more than one plausible option. For instance, if the action of is not free, then , a Deligne–Mumford stack, might serve just as well as the scheme or algebraic space . The difference between these alternatives becomes particularly striking when acts trivially on — the quotient can be regarded as a degenerate case of a double covering, ramified everywhere, whereas is a -Galois covering, ramified nowhere. From the point of view of the first quotient, all involutions will appear to be of the first kind, whereas with respect to the second quotient, all involutions will appear to be of the second kind.
We are therefore led to conclude that a chosen quotient , in addition to and , is necessary in order to discuss involutions in a way consistent with what is already done in the cases where is an involution of a ring.
We require a quotient to satisfy certain axioms, presented in Subsection 4.3, and prove that they are satisfied in a number of important examples, notably when the categorical quotient exists in the category of schemes and is a good quotient. Such quotients exist for instance if is affine or projective, see Theorem 4.4.4. Thereafter in the development of the theory, we are usually agnostic about the quotient chosen. In examples, we often return to the motivating case of a good quotient.
Consider, therefore, a good quotient . It is technically easier to work on than on . Specifically, by virtue of our Theorem 4.3.11, there is an equivalence between Azumaya algebras with -involution on on the one hand and Azumaya algebras with -involution over the sheaf of rings on the other. We therefore study Azumaya algebras over . While does not carry an involution, the ring sheaf has an involution, namely, , which we abbreviate to . A difficulty that we encounter here is that the sheaf of rings is not a local ring object on , but rather a sheaf of rings with involution, the fixed subsheaf of which is the local ring object . We devote considerable work to the study of commutative rings with involutions whose fixed subrings are local in Section 3, and conclude in Theorem 3.3.8 that any such ring is a semilocal ring, so that the sheaf may be viewed as making a “semilocally ringed” space.
In Section 5, we introduce and study types of -involutions. Specifically, we define two Azumaya -algebras with a -involution, and , to be of the same type if some matrix algebra over is -locally isomorphic to some matrix algebra over . We show in Theorem 5.2.13 and Corollary 5.2.14 that the collection of types forms a -torsion group whose product rule is compatible with tensor products, and when , the involutions and have the same type if and only if and are -locally isomorphic, without the need to pass to matrix algebras. Thus, the definition given here agrees with the definition in Subsection 1.1. We then turn to the problem of calculating the group of types in specific cases.
Let denote the branch locus of . Then, away from , the -action on is unramified, hence there is only one possible type of -involution on , viz. unitary, and all involutions on are locally isomorphic to the involution given by applying the involution to each entry in the matrix and then taking the transpose, i.e., . In contrast, over a connected component of , regarded as a reduced closed subscheme of , the involution restricts to the identity (Proposition 4.5.5), and so -involutions of fall into one of two types — orthogonal or symplectic. This suggests that the types of -involutions over should be in bijection with , where and and represent orthogonal and symplectic involutions respectively, and that two -involutions are of the same type if and only if they are of the same type when restricted to each connected component of . We prove the second statement in Theorem 5.4.5 and establish the first under the assumption that that is noetherian and regular in Corollary 6.4.8. We do not know whether the first statement holds in general. Determining the type of a given involution of a given algebra, , can now be carried out by considering the rank of the sheaf of -symmetric elements on the various components of ; see [knus_book_1998-1, Prp. 2.6].
In Section 6, we turn to the question of when a Brauer class contains an algebra possessing a -involution. Saltman [saltman_azumaya_1978, Thm. 3.1] gave necessary and sufficient conditions for this when is trivial or unramified. Specifically, is equivalent to such an algebra if , in the case of a trivial action, or if , in the case of an unramified action. We unify these two results, and generalize to the cases that are neither trivial nor unramified, by defining a transfer map , and deducing in Theorem 6.3.3 that is equivalent to an algebra admitting a -involution of type if and only if , where is a cohomology class depending on the type. In both extreme cases of trivial and unramified actions, and in fact whenever is a nonsingular variety, is necessarily . Moreover, in the case of a trivial action, , and in the unramified case, , so we recover Saltman’s theorem as a special case. We also show that if is equivalent to an algebra with involution, then such an algebra can be constructed to have degree twice that of , thus extending the analogous result of [knus_azumaya_1990, Thms. 4.1, 4.2]. We do not, however, follow [saltman_azumaya_1978] and [knus_azumaya_1990] in considering the corestriction algebra of , taking instead a purely cohomological approach. In fact, it is not clear whether a corestriction algebra of can be defined in a meaningful way when is ramified. This problem was considered in [auel_parimala_suresh_2015, §5], where some positive results are given, and we leave its pursuit in the current level of generality to a future work.
Section 7 gives a number of examples of the workings out of the previous theory. In particular, we give examples of schemes with involutions that are neither unramified nor trivial, along with a classification of the various types of -involutions of Azumaya algebras, e.g., Examples 7.1.6 and 7.1.7.
While this overview has so far been written in the language of schemes, the majority of the results are established in the setting of locally ringed Grothendieck topoi, of which the étale ringed topoi of a scheme is a special case. The advantage of this generality is that all the results above also apply, essentially verbatim, to Azumaya algebras with involution over a topological -space, or to Azumaya algebras with involutions on algebraic stacks. The applicability of our results in the context of other sites associated with schemes, e.g., the Zariski site, the fppf site, the Nisnevich site and some large sites, is discussed in Subsection 4.4.
Comparison of Azumaya algebras over schemes with topological Azumaya algebras has proved useful in the past, for instance in [antieau_unramified_2014], [antieau_topology_2015]. Having the previous theory available also in the topological context, we consider a finite type, regular -algebra with an unramified involution and compare the theory of Azumaya -algebras with involutions restricting to on the centre with the theory of topological Azumaya algebras with involution on the complex manifold . This is carried out in Subsection 4.2, specifically in Example 4.2.4.
By such comparison, we produce an example of an Azumaya algebra of degree , over a ring with an unramified involution , having the property that is Brauer equivalent to an algebra with -involution, but the least degree of such an is , Theorem 9.2.3; the bound is the lowest possible by [knus_azumaya_1990, §4], which guarantees the existence of of degree in general. An analogous example in the case where is assumed to be trivial was given in [asher_auel_azumaya_2017]. The method of proof, which is carried out in Sections 8 and 9, is by using existing study of bundles with involution as a branch of equivariant homotopy theory, [may_equivariant_1996]. In particular, we can find universal examples of topological Azumaya algebras with involution, which are valuable sources of counterexamples.
In an appendix, we give a proof that the stalks of the sheaf of continuous, complex-valued functions on a topological space satisfy Hensel’s lemma. This is used here and there in the body of the paper to treat this case at the same time as étale sites of schemes.
1.3. Acknowledgments
The authors would like to thank Zinovy Reichstein for introducing them to each other and recommending that they study involutions of Azumaya algebras from a topological point of view. They would like to thank Asher Auel for helpful conversations and good ideas, some of which appear in this paper. They owe an early form of an argument in 5.4 to Sune Precht Reeh. The second author would like to thank Omar Antolín, Akhil Mathew, Mona Merling, Marc Stephan and Ric Wade for various conversations about equivariant classifying spaces, and Bert Guillou for a reference to the literature on equivariant model structures. The second author would like to thank Ben Antieau for innumerable valuable conversations about Azumaya algebras from the topological point of view, and would like to thank Gwendolyn Billett for help in deciphering [giraud_cohomologie_1971]. We also thank the referees for many valuable suggestions.
2. Preliminaries
This section recalls necessary facts and sets notation for the sequel. Throughout, denotes a Grothendieck topos. We reserve the term “ring” for commutative unital rings, whereas algebras are assumed unital but not necessarily commutative.
2.1. Generalities on Topoi
Recall that a Grothendieck topos is a category that is equivalent to the category of set-valued sheaves over a small site, or equivalently, a category satisfying Giraud’s axioms; see [giraud_cohomologie_1971, Chap. 0]. In this paper we shall be particularly interested in the following examples:
-
(i)
, the category of sheaves over the small étale site of a scheme .
-
(ii)
, the category of sheaves on a topological space .
We will occasionally consider other sites associated with a scheme . In particular, and will denote the small Zariski and small fppf sites of , respectively.
The topos of sheaves over a singleton topological space, which is nothing but the category of sets, will be denoted .
We note that every topos can be regarded as a site relative to its canonical topology. In this case, a collection of morphisms is a covering of if and only if it is jointly surjective, and every sheaf over is representable, so that . This allows us to define objects of by specifying the sheaf that they represent, and to define morphisms between objects by defining them on sections.
The symbols and will be used for the initial and final objects of , respectively. When for a site , the sheaf assigns an empty set to every non-initial object of , and is the sheaf assigning a singleton to every object in . The subscript will dropped when it may be understood from the context.
For every object of the -sections of are
and the global sections of are . We will write , and will regard as an object of the slice category .
By a group in we will mean a group object in . In this case, the -sections form a group for all objects of . Similar conventions will apply to abelian groups, rings, -objects, and so on.
If is a ring object in some topos, then will denote the object of square-roots of in , that is, the object given section-wise by . The bald notation will denote the constant sheaf .
2.2. Torsors
Definition 2.2.1.
Let be a site and let be a sheaf of groups on . A (right) -torsor is a sheaf on equipped with a right action such that is locally isomorphic to as a right -object.
Equivalently, and intrinsically to the topos , a (right) -torsor is an object of equipped with a (right) -action such that the unique morphism is an epimorphism and such that the morphism is an isomorphism. See [giraud_cohomologie_1971, Déf. III.1.4.1] where more general torsors over objects of are defined; our definition is that of torsors over the terminal object.
The equivalence of the two definitions of “torsor” is given by [giraud_cohomologie_1971, Prop. III.1.7.3].
The category of -torsors, with -equivariant isomorphisms as morphisms, will be denoted
A -torsor is trivial if as right -objects, and an object is said to trivialize if as -objects. The latter holds precisely when .
Recall that if is a -torsor and is a left -object in , then denotes the quotient of by the equivalence relation given by on sections. We shall sometimes denote by and call it the -twist of . We remark that and are locally isomorphic in the sense that there exists a covering in such that — take any such that . If posses some additional structure, for instance if is an abelian group, and respects this structure, then also posses the same structure and the isomorphism respects the additional structure. The general theory outlined here is established precisely in [giraud_cohomologie_1971, Chap. III].
Remark 2.2.2.
There is another plausible definition of “torsor” on a site , particularly when the topology is subcanonical and when the category has finite products—i.e., is a standard site. That is, one modifies the definition in 2.2.1 by requiring the objects and to be objects of the site . These are the representable torsors as distinct from the sheaf torsors defined above. We will not consider the question of representability in this paper beyond the following remark: Suppose is a scheme and is a group scheme over . Then represents a group sheaf on the big flat site of , also denoted . If is affine, then all sheaf -torsors are representable by an -scheme [milne_etale_1980, Thm. III.4.3].
2.3. Cohomology of Abelian Groups
The functor sending an abelian group in to its global sections is left exact. The -th right derived functor of is denoted , as usual. If is clear from the context, we shall simply write . When for a scheme , we write as , and likewise for other sites associated with .
In the sequel, we shall make repeated use of Verdier’s Theorem, quoted below, which provides a description of cohomology classes in terms of hypercoverings. We recall some details, and in doing so, we set notation. One may additionally consult [de_jong_stacks_2017, Tag 01FX], [dugger_hypercovers_2004] or [artin_theorie_1972, Exp. V.7].
Let denote the category having as its objects and the non-decreasing functions as its morphisms. Recall that a simplicial object in is a contravariant functor . For every , we write and set and , where is the non-decreasing monomorphism whose image does not include and is the non-decreasing epimorphism for which has two preimages. We shall write instead of when is clear from the context. Since the morphisms generate , in order to specify a simplicial object in , it is enough to specify objects and morphisms , for all . Of course, the morphisms have to satisfy certain relations, which can be found in [may_simplicial_1992], for instance.
For , let denote the full subcategory of whose objects are . The restriction functor admits a right adjoint called the -th coskeleton and denoted . We also write for . The simplicial object is called a hypercovering (of the terminal object) if is a covering and for all , the map induced by the adjunction is a covering. For example, when , the latter conditions means that is a covering.
Hypercoverings form a category in the obvious manner, morphisms being natural transformations.
Example 2.3.1.
Let be a morphism in . Define ( times), let be the projection omitting the -th copy of and let be given by on sections. These data determine a simplicial object which is a hypercovering if is a covering. In this case, the map is an isomorphism for all . The hypercovering is called the Čech hypercovering associated to . If is an arbitrary hypercovering, then is the Čech hypercovering associated to .
The following lemma is fundamental.
Lemma 2.3.2 ([de_jong_stacks_2017, Lm. 24.7.3] or [artin_theorie_1972, Th. V.7.3.2]).
Let be a hypercovering and let be a covering. Then there exists a hypercovering morphism such that factors through .
Let be an abelian group object of . With any hypercovering in we associate a cochain complex defined by for and otherwise. The coboundary map is given by , as usual; here is the map induced by . The cocycles, coboundaries, and cohomology groups of the complex are denoted , and . Any morphism of hypercoverings induces a morphism in the obvious manner.
Theorem 2.3.3 (Verdier [artin_theorie_1972, Th. V.7.4.1]).
Let be a topos and an abelian group object in . The functors
from the category of abelian groups in to the category of abelian groups are naturally isomorphic. Here, the colimit is taken over the category of hypercoverings.
Remark 2.3.4.
If we were to take the colimit in the theorem over the category of the Čech hypercoverings, then the result would be the Čech cohomology of . Consequently, the Čech cohomology and the derived-functor cohomology agree when every hypercovering admits a map from a Čech hypercovering. This is known to be the case when for a paracompact Hausdorff topological space [godement_topologie_1973, Th. 5.10.1], or for a noetherian scheme such that any finite subset of is contained in an open affine subscheme [artin_joins_1971, §4].
A short exact sequence of abelian groups in gives rise to a long exact sequence of cohomology groups. By the second proof of [de_jong_stacks_2017, Tag 01H0], quoted as Theorem 2.3.3 here, the connecting homomorphism can be described as follows: Let be a cohomology class represented by a cocycle for some hypercovering . Since is an epimorphism, we may find a covering such that is the image of some . By Lemma 2.3.2 there exists a morphism of hypercoverings such that factors through . We replace with its image in . One easily checks that the image of in both and is , and hence . Now, is the cohomology class determined by .
2.4. Cohomology of Non-Abelian Groups
For a group object of , not necessary abelian, we define the pointed set by hypercoverings. Given a hypercovering in , let be the set of elements satisfying
| (2) |
in ; here is induced by . Two elements are said to be cohomologous, denoted , if there exists such that . We define the pointed set to be with the equivalence class of as a distinguished element. A morphism of hypercoverings induces a morphism of pointed sets . Now, following the literature, we define
where the colimit is taken over the category of all hypercoverings in . We note that some texts take the colimit over the category of Čech hypercoverings, see Example 2.3.1, but this makes no difference thanks to the following lemma.
Lemma 2.4.1.
Let be a hypercovering. Then the maps and , induced by the canonical morphism , are isomorphisms.
Proof.
The proof shall require various facts about coskeleta. We refer the reader to [de_jong_stacks_2017, §14.19] or any equivalent source for proofs.
Recall from Example 2.3.1 that is nothing but the Čech hypercovering associated to . Since is a hypercovering, is a covering, and hence the induced map is injective. Since the map is a restriction of the latter, it is also injective. This implies that if two cocycles in become cohomologous in , then they are also cohomologous in , so is injective. It is therefore enough to show that is surjective.
We first observe that the canonical map is an isomorphism. This follows from the fact that and is a covering, hence (2) is satisfied in if and only if it is satisfied in . Since , we may replace with . In this case, the construction of implies that is characterized by
for all objects of . The maps are then given by taking the -part, -part, and -part, respectively. Geometrically, is the object of simplicial morphisms from the boundary of the -simplex to .
Let . We claim that descends along to . Write and let , denote the first and second projections from onto . We need to show that in . For an object , the -sections of can be described by
Define by
on sections. One readily checks that , and . Now, applying to (2), we arrive at the equation in , and applying to (2), we find that in . Both equations taken together imply that in , hence our claim follows.
To finish the proof, it is enough to show that is a -cocycle. This will follow from the fact that is a -cocycle if we show that the canonical map is a covering. The latter map is given by on sections. Since is a covering, for every pair of vertices , there exists a covering and an edge satisfying , . This easily implies that is locally surjective, finishing the proof. ∎
The following proposition summarizes the main properties of . As before, we shall suppress , writing , when it is clear form the context.
Proposition 2.4.2.
Let be a short exact sequence of groups in .
-
(i)
is naturally isomorphic to the set of isomorphism classes of -torsors; the distinguished element of corresponds to the isomorphism class of the trivial torsor.
-
(ii)
There is a long exact sequence of pointed sets
This exact sequence is functorial in , i.e., a morphism from it to another short exact sequence of groups gives rise a morphism between the corresponding long exact sequences.
-
(iii)
When is central in , one can extend the exact sequence of (ii) with an additional morphism , which is again functorial in .
- (iv)
The proposition is well known, but Giraud [giraud_cohomologie_1971, IV, 4.2.7.4, 4.2.10] is the only source we are aware of that treats all parts in the generality that we require. Since the treatment in [giraud_cohomologie_1971] is somewhat obscure, and since we shall need the definition of the maps and in the sequel, we include an outline of the proof here. Note that it is easier to prove (i) using the definition of via Čech hypercoverings, while it is easier to prove (iii) using the definition of via arbitrary hypercoverings, and these definitions are equivalent thanks to Lemma 2.4.1.
Proof (sketch).
-
(i)
Let be a -torsor. Choose a covering such that and fix some . Form the Čech hypercovering associated to . Then there exists a unique such that in . We leave it to the reader to check that and the construction induces a well-defined map from the isomorphism classes of to taking the trivial -torsor to the special element of .
In the other direction, let . By Lemma 2.4.1, is represented by some where is a Čech hypercovering. Define to be the object of characterized by ; here, is induced by . There is a right -action on given by on sections. We leave it to the reader to check that is indeed a -torsor, and the assignment defines an inverse to the map of the previous paragraph. In particular, note that is a covering because ; use the fact that is a Čech hypercovering.
-
(ii)
Define as follows: Let . There is a covering such that lifts to some . One easily checks that lies in where is the Čech hypercovering associated to . We define to be the cohomology class represented by , and leave it to the reader to check that this is well-defined. All other maps in the sequence are defined in the obvious manner and the exactness is easy to check.
-
(iii)
Define as follows: Let be represented by where is a hypercovering. There is a covering such that lifts to some . By Lemma 2.3.2, there is a morphism of hypercoverings such that factors through . We replace with its image in . Let . It is easy to check that lies in and defines a -cocycle of relative to . We define to be the cohomology class represented by , and leave it to the reader to check that this is well-defined. The exactness of the sequence at is straightforward to check.
- (iv)
2.5. Azumaya Algebras
Let be a ring object of and let be a positive integer. Recall that an Azumaya -algebra of degree is an -algebra in that is locally isomorphic to , i.e., there exists a covering such that as -algebras. The Azumaya -algebras of degree together with -algebra isomorphisms form a category which we denote by
If is another Azumaya -algebra, we let denote the subobject of the internal mapping object of consisting of -algebra homomorphisms. We define the group object similarly.
Remark 2.5.1.
We have defined here Azumaya algebras of constant degree only. When is connected, these are all the Azumaya algebras, but in general, one has to allow the degree to take values in the global sections of the sheaf of positive integers on . For any such one can define and the definition of Azumaya algebras of degree extends verbatim. We ignore this technicality, both for the sake of simplicity, and also since it is unnecessary for connected topoi, which are the topoi of interest to us.
Let be a ring object in . Recall that is locally ringed by , or is a local ring object in , if for any object in and with , there exists a covering such that for all . In fact, one can take for almost all . We remark that the condition should also hold when , which implies that cannot be the zero ring when . When has enough points, the condition is equivalent to saying that for every point , the ring is local (the zero ring is not considered local).
Suppose that is a local ring object. Then the group homomorphism given by on sections is surjective, [giraud_cohomologie_1971, V.§4]. This induces an isomorphism , which will be used to freely identify the source and target in the sequel. The following proposition is well established, again see [giraud_cohomologie_1971, V.§4].
Proposition 2.5.2.
If is a local ring object, then there is an equivalence of categories
given by the functors and .
The proposition holds for any ring object of if one replaces the group object with .
We continue to assume that is a local ring object. By Proposition 2.4.2(iii), the short exact sequence gives rise to a pointed set map ; here, . As usual, the Brauer group of is
the addition being that inherited from the group . Since Azumaya -algebras correspond to -torsors, which are in turn classified by , any Azumaya -algebra gives rise to an element in , denoted and called the Brauer class of . By writing or saying that is Brauer equivalent to , we mean that is an Azumaya -algebra with . For more details, see [grothendieck_groupe_1968-1] or [giraud_cohomologie_1971, Chap. V, §4].
Example 2.5.3.
Let be a topological space, let and let be the sheaf of continuous functions from to , denoted . Then Azumaya -algebra are topological Azumaya algebras over as studied in [antieau_period-index_2014].
Example 2.5.4.
An Azumaya algebra of degree over a scheme is a sheaf of -algebras that is locally, in the étale topology, isomorphic as an -algebra to , [grothendieck_groupe_1968-1, Para. 1.2]
Example 2.5.5.
Let be a ring. An Azumaya -algebra of degree is an -algebra for which there exists a faithfully flat étale -algebra such that as -algebras. This is equivalent to the definition of Example 2.5.4 in the case where by [grothendieck_groupe_1968-1, Th. 5.1, Cor. 5.2]. Consult [knus_quadratic_1991, III.§5] for other equivalent definitions and cf. Remark 2.5.1.
3. Rings with Involution
In this section we collect a number of results regarding involutions of rings that will be needed later in the paper. The main result is Theorem 3.3.8, which gives the structure of those rings with involution for which the fixed ring of is local. It is shown that in this case, is a local ring in its own right, or is a quadratic étale algebra over the fixed ring of . In particular, the ring is semilocal.
Throughout, involutions will be written exponentially and the Jacobson radical of a ring will be denoted .
3.1. Quadratic Étale Algebras
Definition 3.1.1.
Let be a ring. A commutative -algebra is said to be finite étale of rank if is a locally free -module of rank , and the multiplication map may be split as a morphism of -modules, where is regarded as an -algebra via . Finite étale -algebras of rank will be called quadratic étale algebras.
Remark 3.1.2.
One common definition of étale for commutative -algebras is that should be flat over , of finite presentation as an -algebra, and unramified in the sense that , the module of Kähler differentials, vanishes. This is the definition in [grothendieck_elements_1967, Sec. 17.6] in the affine case. Our finite étale algebras of rank are precisely the étale algebras that are locally free of rank .
Indeed, if is locally free of rank over , then is also finitely presented and flat as an -module [de_jong_stacks_2017, Tag 00NX], and hence of finite presentation as an -algebra [grothendieck_elements_1964, Prop. 1.4.7]. Furthermore, admits a splitting if and only if the -ideal is generated by an idempotent, namely . For finitely generated -algebras , the existence of such an idempotent is equivalent to saying is unramified over by [de_jong_stacks_2017, Tag 02FL].
Example 3.1.3.
Let be a monic polynomial of degree . It is well known that is a finite étale -algebra of rank if and only if the discriminant of is invertible in . In particular, is a quadratic étale -algebra if and only if .
Every quadratic étale -algebra admits a canonical -linear involution given by , see [knus_quadratic_1991, I.§1.3.6]. The fixed ring of is and is the only -automorphism of with this property. Moreover, when is connected, it is the only non-trivial -automorphism of .
Proposition 3.1.4.
Let be a local ring with maximal ideal and residue field , let be a quadratic étale -algebra, and let be the unique non-trivial -automorphism of . Then:
-
(i)
-
(ii)
is either a separable quadratic field extension of , or . The automorphism that induces on is the unique non-trivial -automorphism of .
-
(iii)
(“Hilbert 90”) For every with , there exists such that .
Proof.
It is clear that is a quadratic étale -algebra, and hence a product of separable field extensions of [demeyer_separable_1971, Th. II.2.5]. This implies the first assertion of (ii) as well as . The inclusion holds because is a finite -module [reiner_maximal_1975-1, Th. 6.15], so we have proved (i). The last assertion of (ii) follows from the fact that is given by .
Let denote the image of in . To prove (iii), we first claim that there is with . This is easy to see if . Otherwise, is a field and such exists unless for all . The latter forces (take ) and , which is impossible by (ii), so exists.
Let be a lift of . Then is a lift of , which implies by (i). Since , it is the case that , and so with . ∎
3.2. Quadratic Étale Algebras in Topoi
Our definition of a “quadratic étale algebra” extends directly to the case where is a local ring object in a topos .
Definition 3.2.1.
Given a ring object in a topos , we say an -algebra is a finite étale -algebra of rank if is a locally free -module of rank such that the multiplication map may be split as a morphism of -algebras. Finite étale -algebras of rank will be called quadratic étale algebras.
We alert the reader that if is a quadratic étale -algebra, then it is not true in general that is a quadratic étale -algebra for all objects of . In fact may not be locally free of rank over . Rather, one can always find a covering such that is a quadratic étale -algebra; for instance, one may take any such that is a free -module of rank . We further note that in general there is no covering such that as -algebras, e.g. let (the topos of sets) and take and to be and respectively. While is easily seen to be split, there is no covering such that .
As one might expect, being a finite étale algebra of rank is a local property in that it may be tested on a covering.
Lemma 3.2.2.
Let be a ring in , let be an -algebra and let be a covering. Then is a finite étale -algebra of rank if and only if is a finite étale -algebra of rank in .
Proof.
Write . The only non-trivial thing to check is that if the multiplication map admits a splitting in , then so does in . Let denote the first and second projections, and let denote the pullback of along . We claim that admits at most one splitting. Provided this holds, we must have and so descends to a map splitting as required.
The claim can be verified on the level of sections, namely, it is enough to check that any ring surjection admits at most one -linear splitting. If is such a splitting and , then for all , so is determined by the idempotent . It is easy to check that and that is the only idempotent with this property, hence is determined by . ∎
Example 3.2.3.
Let be a quadratic étale morphism of schemes. That is, is affine and can be covered by open affine subschemes such that the ring map corresponding to is quadratic étale for all . Then is a quadratic étale -algebra in both and ; this can be checked using Lemma 3.2.2.
Example 3.2.4.
Let be a double covering of topological spaces and let and denote the sheaves of continuous -values functions on and , respectively. Then is a quadratic étale -algebra in ; again this can be checked with Lemma 3.2.2.
3.3. Rings with Involution
Throughout, is an ordinary commutative ring, is an involution, and is the fixed ring of . The purpose of this section is twofold. First, we show that the locus of primes such that is a quadratic étale over is open in . Second, we study the structure of when is local, showing, in particular, that is quadratic étale over , or is local.
There are two pitfalls in the study of over . First of all, may not be finite over .
Example 3.3.1.
Let be any set, let be the commutative -algebra freely generated by , and let be the -linear involution sending each to . Then the fixed ring of is . Let . Since and , it follows that cannot be generated by fewer than elements as an -algebra. Thus, when is infinite, is not finite over . The same applies to the -algebra , even though is noetherian. We further note that when , the ring is a smooth affine -algebra, but is singular.
Second, the formation of fixed rings may not commute with extension of scalars. That is, if is a commutative -algebra, then need not be the subring of -fixed elements in . In fact, is a priori not one-to-one. Nevertheless, restricts to an isomorphism if is flat over , or . To see this, consider the exact sequence of -modules . The statement amounts to showing that it remains exact after tensoring with . This is clear if is flat, and if , then it follows because is split by .
Remark 3.3.2.
Voight [voight_2011_standard_involution, Corollary 3.2] showed that if is locally free of rank at least over and is not a zerodivisor in , then decomposes as where is an ideal of such that and . Voight calls such (commutative) rings with involution exceptional. This shows that if is not exceptional then either is not locally free over , or . The case and featuring in Example 3.3.1 is an example of an exceptional ring with involution, and essentially the only one if .
Having warned the reader of these pitfalls, we return to the main topic of the section, which is the study of over .
Lemma 3.3.3.
Assume that there exists with . Then is a quadratic étale -algebra.
Proof.
For , write and , and observe that and .
Suppose that . Since , it follows that . Furthermore, if for , then , so because . It follows that the -algebra map sending to is an isomorphism. Since , we conclude that is a quadratic étale -algebra.
We now show that . Write and . One verifies that , so that . Since , we have . Let and . Straightforward computation shows that , hence and . It follows that and thus, . ∎
Lemma 3.3.4.
Suppose that is local and is a quadratic étale -algebra. Then there exists such that .
Proof.
Corollary 3.3.5.
The set of prime ideals such that is a quadratic étale -algebra is open in . Equivalently for every such that is a quadratic étale -algebra, there exists such that is a quadratic étale -algebra.
Proof.
Remark 3.3.6.
We momentarily consider an arbitrary finite group acting on .
Proposition 3.3.7.
Let be a finite group acting on a ring and let be the subring of elements fixed under . If is local then the maximal ideals of form a single -orbit. In particular, is semilocal.
Proof.
Let denote the maximal ideal of and, for the sake of contradiction, suppose and are maximal ideals of lying in distinct -orbits. Let and . Since for all , we have . Using the Chinese Remainder Theorem, choose such that . Replacing with , we may assume that . Since and are both contained in , this means that lies both in and in , which is absurd. ∎
We derive the main result of this section by specializing Proposition 3.3.7 to the case of a group with elements.
Theorem 3.3.8.
Suppose is a ring and is an involution with fixed ring such that is local. Let denote and the restriction of to .
-
(i)
If , then is a quadratic étale algebra over .
-
(ii)
If , then is a local ring that is not quadratic étale over .
In either case, is semilocal.
Proof.
Let be a maximal ideal of . Taking in Proposition 3.3.7, we see that the maximal ideals of are . We consider the cases and separately.
Suppose that . By the Chinese Remainder Theorem, , and under this isomorphism, acts by sending to . This implies that , so we are in the situation of (i). Furthermore, by taking and , we see that there exists such that , or equivalently, . Thus, by Lemma 3.3.3, is quadratic étale over .
Suppose now that . Then is local and is a field. If , then there exists with and again we find that is quadratic étale over . On the other hand, if , then cannot be quadratic étale over by Proposition 3.1.4(ii).
We have verified (i) and (ii) in both cases, so the proof is complete. ∎
Notation 3.3.9.
A henselian ring is a local ring in which Hensel’s lemma, [eisenbud_commutative_1995, Thm. 7.3], holds. A strictly henselian ring is a henselian ring for which the residue field is separably closed.
Lemma 3.3.10.
Let be a finite group acting on a ring and let be the subring of fixed under . If is local with maximal ideal , then . In particular, .
Proof.
Let . To prove , we need to show that , or equivalently, that consists of invertible elements. Let and let denote the distinct elements of . Then
where denotes the -th elementary symmetric polynomial on letters. Since , and since is invariant under , the right hand side lies in . Thus, . ∎
Corollary 3.3.11.
Let be a ring, let be an involution, and let denote the fixed ring of . Suppose that is a strictly henselian ring with maximal ideal . Then is a finite product of strictly henselian rings.
Proof.
By Theorem 3.3.8, either is a quadratic étale -algebra, or is local. In the former case, is finite over , and the lemma follows from [de_jong_stacks_2017, Tag 04GH] so we assume is local. Write for the maximal ideal of , for its residue field, and denote by the surjection .
Observe first that is integral over since for all , we have and . This implies that is algebraic over the residue field of , which is separably closed, hence is separably closed.
Now, let be a monic polynomial such that has a simple root . We will show that has root with . Let be the coefficients of and let be any element with . Since is integral over , there is a finite -subalgebra containing . By [de_jong_stacks_2017, Tag 04GH], is a product of henselian rings. Since has no non-trivial idempotents, this means is a henselian ring. Write for the maximal ideal of and . Since is finite over , there is a natural number such that by [reiner_maximal_1975-1, Th. 6.15]. This implies that and the latter is a proper ideal of , by Lemma 3.3.10. Therefore, and we have a well-defined homomorphism of fields given by . Since , we have and . As is a henselian ring, there is with and . This completes the proof. ∎
4. Ringed Topoi with Involution
Unless indicated otherwise, throughout this section, denotes a locally ringed topos. Our interest is in the following examples:
-
(1)
for a scheme , and is the structure sheaf of which sends to .
-
(2)
for a topological space , and is the sheaf of continuous -valued functions, denoted .
We write the cyclic group with two elements as and, when applicable, the non-trivial element of will always be denoted .
When there is no risk of confusion, we shall refer to as a ringed topos, in which case the ring object is understood to be .
4.1. Involutions of Ringed Topoi
Definition 4.1.1.
Suppose is a topos. An involution of consists of an equivalence of categories and a natural isomorphism satisfying the coherence condition that for all objects of .
The natural equivalence will generally be suppressed from the notation.
Definition 4.1.2.
Suppose is a ringed topos. An involution of consists of an involution of and an isomorphism of ring objects such that .
Suppressing from the last equation, we say .
Remark 4.1.3.
(i) The functor is left adjoint to itself with the unit and counit of the adjunction being and , respectively. Thus, if we write , then the adjoint pair defines a geometric automorphism of . Moreover, defines an automorphism of the ringed topos .
(ii) Topoi and ringed topoi form -categories in which there is a notion of a weak -action. An involution of a topos, resp. ringed topos, as defined here induces such a weak -action in which the non-trivial element acts as the morphism , resp. , and the trivial element acts as the identity. It can be checked that all -actions with the latter property arise in this manner. Since an arbitrary weak -action is equivalent to one in which acts as the identity, specifying an involution is essentially the same as specifying a weak -action.
Notation 4.1.4.
Henceforth, when there is no risk of confusion, involutions of ringed topoi will always be denoted . In fact, we shall often abbreviate the triple to .
It is convenient to think of as the “geometric data” of the involution and of as the “arithmetic data” of the involution. The following are the motivating examples.
Example 4.1.5.
Let be a scheme and let be an involution. The direct image functor defines an involution of ; the suppressed natural isomorphism is the identity. Let be the structure sheaf of on . The involution of gives rise to an isomorphism as follows: For an étale morphism , define via the pullback diagram
By definition, , and we define to be the isomorphism induced by . It is easy to check that and so is an involution of . When for a ring and is induced by an involution of , we can recover that involution from by taking global sections.
The small étale site can be replaced by other sites, for instance the site .
Example 4.1.6.
Let be topological space with a continuous involution . Write and let , the ring sheaf of continuous functions into . Then the direct image functor defines an involution of ; the isomorphism is the identity.
In particular, . Let be given sectionwise by precomposition with , namely
Then is an involution of .
Example 4.1.7.
Let be any topos, let be a ring object in and let be an involution. Then is an involution of .
Definition 4.1.8.
The trivial involution of is . Any other involution of will be said to be weakly trivial if it is equivalent to the trivial involution of in the following sense: There exists a natural isomorphism such that and for all objects in .
Example 4.1.9.
Let be an involution of and let be an -module. Then carries a -module structure. We shall always regard as an -module by using the morphism , that is, by twisting the structure.
Using the equality , one easily checks that is an isomorphism of -modules, which we suppress from the notation henceforth.
Notice that contrary to the case of -twisting of modules over ordinary rings, it is not in general true that can be identified with its -twist as an abelian group object in . For instance, in the context of Example 4.1.5, suppose that and exchanges two maximal ideals , and let , denote the quasicoherent -modules corresponding to the -modules and , respectively. Then , but is not isomorphic to as abelian group objects in because is supported at while is supported at .
If is an -algebra, then , besides being an -module, carries an -algebra structure. Letting denote the opposite algebra of , we have up to the suppressed natural isomorphism .
Definition 4.1.10.
A -involution on an -algebra is a morphism of -algebras such that . In this case, is called an -algebra with a -involution. If is an Azumaya algebra, it will be called an Azumaya algebra with -involution.
If and are -algebras with -involutions, then a morphism from to is a morphism of -algebras such that .
Notice that applying to both sides of gives .
Notation 4.1.11.
The category where the objects are degree- Azumaya -algebras with -involutions and where the morphisms are isomorphisms of -algebras with -involutions shall be denoted , or just .
Given a ring with involution and an -algebra , it is reasonable to define a -involution of to be an involution satisfying for all , . Following Gille [gille_gersten_2009, §1], this can be generalized to the context of schemes: Given a scheme , an involution , and an -algebra , then a -involution of is an -algebra morphism such that . We reconcile these elementary definitions with Definition 4.1.10 in the following example.
Example 4.1.12.
Let be a ring with involution, let and write for the involution induced by . Abusing the notation, let denote the involution induced by on the étale ringed topos of , see Example 4.1.5.
Every -module gives rise to an -module, also denoted , the sections of which are given by for any in . In fact, this defines an equivalence of categories between -modules and quasicoherent -modules [de_jong_stacks_2017, Tag 03DX]. Straightforward computation shows that the -module corresponds to , the -module obtained from by twisting via .
Now let be an -algebra and let be an involution satisfying for all , . Realizing as an -algebra in , the algebra corresponds to the -algebra , and so the involution induces a -involution , also denoted . By taking global sections, we see that all -involutions of the -algebra are obtained in this manner.
Likewise, if is a scheme, is an involution, and is a quasicoherent -algebra in , then the -involutions of the -algebra associated to in (see [de_jong_stacks_2017, Tag 03DU]) are in one-to-one correspondence with the -involutions of in the sense of Gille [gille_gersten_2009, §1]. The quasicoherence assumption applies in particular when is an Azumaya algebra over .
Example 4.1.13.
Let be an involution of and let be a natural number. Then admits a -involution given by on sections and denoted . If the sections lie in , then the sections lie in .
4.2. Morphisms
We now give the general definition of a morphism of ringed topoi with involution. Only very few examples of these will be considered in the sequel.
Recall that a geometric morphism of topoi consists of two functors , together with an adjunction between and and such that commutes with finite limits. We shall usually denote the unit and counit natural transformations associated to the adjunction by and , dropping the superscript when there is no risk of confusion. If and are ringed topoi, then a morphism consists of a geometric morphism of topoi together with a ring homomorphism , which then corresponds to a ring homomorphism via the adjunction.
Now let and be ringed topoi with involutions and , respectively. Regarding and as geometric automorphisms of and , see Remark 4.1.3, a morphism of ringed topoi with involution should intuitively consist of a morphism of ringed topoi such that is “equivalent” to . The specifications of this equivalence, which we now give, are somewhat technical.
Definition 4.2.1.
With the previous notation, a morphism of ringed topoi with involution consists of a morphism of ringed topoi together with natural isomorphisms and satisfying the following coherence conditions for all objects in and in :
-
(1)
The following diagram, the columns of which are induced by and and the rows of which are induced by the relevant adjunctions, commutes.
-
(2)
-
(3)
-
(4)
-
(5)
We say that is strict when and , so that and .
We call an equivalence when is an equivalence of categories and is an isomorphism, in which case the same holds for and .
Remark 4.2.2.
Yoneda’s lemma and condition (1) imply that and determine each other. Explicitly, this is given as
Furthermore, provided (1) holds, conditions (2) and (4) are equivalent, and so are (3) and (5). Thus, in practice, it is enough to either specify and verify (2) and (3), or specify and verify (4) and (5).
In accordance with Remark 4.1.3, we will sometimes call morphisms of ringed topoi with involution -equivariant morphisms.
We will usually suppress and in computations, identifying with and with . The coherence conditions guarantee that this will not cause inconsistency or ambiguity.
Example 4.2.3.
Let be a weakly trivial involution of and let be the trivial involution on , see Definition 4.1.8. Then there is such that and for all in , and one can readily verify that defines an equivalence from to , and that every such equivalence is of this form.
More generally, if is arbitrary and there exists an equivalence from to a ringed topos with a trivial involution, then is weakly trivial in the sense of Definition 4.1.8; take .
If is a morphism of ringed topoi with involution, and if is an Azumaya -algebra with a -involution , then is an Azumaya algebra on with a -involution given by . This induces transfer functors
We shall need a particular instance of these transfer maps later.
Example 4.2.4.
If is a complex variety with involution , then the étale ringed topos of , denoted , becomes a locally ringed topos with involution, as in Example 4.1.5. On the other hand, one may form the topological space , equipped with the analytic topology, which then has an associated site and consequently an associated topos . One may endow the topos with several different local ring objects depending on the kind of geometry one wishes to carry out. There is the sheaf of holomorphic functions, as set out in [grothendieck_techniques_1960], and there is also the sheaf of continuous -valued functions.
We claim that is a locally ringed topos and that there is a “realization” morphism of ringed topoi with involution. An outline of the argument follows.
For every complex variety , there is a unique, functorially-defined analytic topology on ; this is established in [grothendieck_revetements_1971, Exp. XII]. The functor preserves finite limits. Moreover, if is an étale covering of , then the family of maps is a jointly surjective family of local homeomorphisms, and may therefore be refined by a jointly surjective family of open inclusions. Since families of this form generate the usual Grothendieck topology on the topological space , it follows that there is a morphism of sites , and therefore a morphism of topoi, [artin_theorie_1972-1, III.1 and IV.4.9.4]. Complex realization may be applied to , the representing object for , to obtain , the representing object for which is the local ring object on . Therefore, is a morphism of locally ringed topoi. It is routine to verify that since the involution on becomes the obvious involution on after realization, the morphism extends to a strict morphism of locally ringed topoi with involution. In this instance, all the “coherence” natural isomorphisms appearing are, in fact, identities.
4.3. Quotients by an Involution
Let be an involution of a locally ringed topos . We would like to consider a quotient of by the action of , or equivalently, by the (weak) -action it induces. In general, however, it is difficult to define a specific quotient topos in a geometrically reasonable way. For example, if for a scheme admitting a -action, then the étale ringed topoi of both the geometric quotient , if exists, and the stack may a priori serve as reasonable quotients of .
We therefore ignore the problem of constructing or specifying a quotient of a locally ringed topos with involution and instead enumerate the properties that such a quotient should possess, declaring any locally ringed topos possessing these properties to be satisfactory.
To be precise, we ask for a locally ringed topos , endowed with the trivial involution, together with a -equivariant morphism which satisfy certain axioms. Recall from Subsection 4.2 that the data of consists of a geometric morphism of topoi , a ring homomorphism (or equivalently, ) and natural transformations , satisfying the relations of Definition 4.2.1. We will often suppress and , identifying with and with . In fact, in many of our examples, and will both be the identity.
Definition 4.3.1.
Let be a locally ringed topos with involution , let be a locally ringed topos with a trivial involution, and let be a -equivariant morphism of ringed topoi. We say that is an exact quotient (of by the given -action) if
-
(E1)
is the equalizer of and the identity map ,
-
(E2)
preserves epimorphisms.
Remark 4.3.2.
An exact quotient is in particular a morphism of locally ringed topoi, i.e., a morphism of ringed topoi satisfying the additional condition that is the pullback of along . Indeed, given , the -sections of the pullback consist of pairs with in . By (E1), we have in . Since , this means that . Thus, again by (E1), there exists unique with . In particular, in . Since is a subobject of via (again, by (E1)), this means that in , so . As , it follows that is the image of a (necessarily unique) -section of under the natural map , as required.
The name “exact” comes from condition (E2), which implies in particular that preserves exact sequences of groups. We shall see below that this condition is critical for transferring cohomological data from to . Condition (E1) informally means that behaves as one would expect from the subring of fixed by — such an object cannot be defined in because the source and target of are not, in general, canonically isomorphic.
To motivate Definition 4.3.1, we now give two fundamental examples of exact quotients. However, in order not to digress, we postpone the proof of their exactness to Subsection 4.4, where further examples and non-examples are exhibited.
Example 4.3.3.
Let be a scheme and let be an involution. A morphism of schemes is called a good quotient of relative to the action of if is affine, -invariant, and defines an isomorphism of with . By [grothendieck_revetements_1971, Prp. V.1.3] and the going-up theorem, these conditions imply that is universally surjective and so this agrees with the more general definition in [de_jong_stacks_2017, Tag 04AB]. A good -quotient of exists if and only if every -orbit in is contained in an affine open subscheme, in which case it is also a categorical quotient in the category of schemes, hence unique up to isomorphism [grothendieck_revetements_1971, Prps. V.1.3, V.1.8].
Let , and let be the involution of induced by , see Example 4.1.5. Given a good -quotient, , we define an exact quotient relative to by taking , letting and defining to be the canonical extension of in to the corresponding ring objects in . The suppressed natural transformations and are both the identity.
Example 4.3.4.
Let be a Hausdorff topological space, let be a continuous involution, let and let be the quotient map. Let and let be the involution induced by , see Example 4.1.6. We define an exact quotient relative to by taking , letting be the geometric morphism induced by , and defining to be the morphism sending a section to . Again, the suppressed natural transformations and are both the identity.
We also record the following trivial example.
Example 4.3.5.
Suppose that the involution on is weakly trivial, namely, there is a natural isomorphism such that , and . Then the identity morphism defines an exact quotient upon taking and . We call it the trivial quotient of .
More generally, an arbitrary exact -quotient will be called trivial when is an equivalence of ringed topoi with involution. As noted in Example 4.2.3, such a quotient can only exist when the involution of is weakly trivial.
We shall see below (Remark 4.5.9) that a locally ringed topos with involution may admit several non-equivalent exact quotients.
We turn to establish some properties of exact quotients that will arise in the sequel. The most crucial of these will be the fact that when is an exact -quotient, induces an equivalence between the Azumaya -algebras and the Azumaya -algebras, and similarly for Azumaya algebras with a -involution.
The following theorem is a consequence of condition (E2).
Theorem 4.3.6.
Let be a geometric morphism of topoi such that preserves epimorphisms, and let be a group in . Then:
-
(i)
induces an equivalence of categories .
- (ii)
-
(iii)
If is a short exact sequence of abelian groups then the isomorphism of (ii) gives rise to an isomorphism between the cohomology long exact sequence of and the cohomology long exact sequence of . The same holds for the truncated long exact sequence of parts (ii) and (iii) of Proposition 2.4.2 when are not assumed to be abelian.
Proof.
-
(i)
We treat the topos as a site in its own canonical topology. In this language, [giraud_cohomologie_1971]*Chap. V, Sect. 3.1.1.1 says that induces an equivalence , where the source is the category of -torsors for which there exists a covering such that . We claim that this applies to all -torsors, and so induces an equivalence .
Since a -torsor is trivialized by itself, it is also trivialized by any object mapping to , for instance by . It is therefore sufficient to show that the map is an epimorphism. Since is an epimorphism, our assumption implies that is also an epimorphism, so the claim is verified.
-
(ii)
Suppose first that is abelian. The fact that is exact implies that the family of functors from abelian groups in to abelian groups forms a -functor. Thus, the universality of derived functors implies that the canonical isomorphism gives rise to a unique natural transformation for any abelian . Since takes injective abelian groups to injective abelian groups, is an effaceable -functor, hence universal. Applying the universality of the latter to the natural isomorphism implies that is an isomorphism.
We note that if we use Verdier’s Theorem, see Subsection 2.3, to describe and , then the isomorphism is given by sending the cohomology class represented by to the cohomology class represented by . Notice that , which is just , is a hypercovering since preserves epimorphisms and commutes with for all . This isomorphism coincides with the one in the previous paragraph because they coincide on the -th cohomology.
- (iii)
Remark 4.3.7.
When is a geometric morphism of topoi, with no further assumptions, one still has natural transformations for all abelian and , or non-abelian and , and if preserves epimorphisms, they coincide with the inverses of the isomorphisms of Theorem 4.3.6. In the abelian case, the construction is given as follows: Using the exactness of , one finds that the canonical map gives rise to natural transformations . Taking and composing with the map , induced by the counit , one obtains a natural transformation . This map can be written explicitly on the level of cocycles, using Verdier’s Theorem, and be adapted to the non-abelian case when .
Henceforth, let be an exact quotient relative to an involution on . We write
for brevity, and, abusing the notation, we let denote the involution .
We shall use the following lemma freely to identify with and with .
Lemma 4.3.8.
For all , there are canonical isomorphisms , and .
Proof.
Let . Thanks to the adjunction between and , we have a natural isomorphism . This establishes the first isomorphism.
The second isomorphism is obtained in the same manner.
The last isomorphism is deduced from the following ladder of short exact sequences
Here, the left and middle isomorphisms follow from the previous paragraph, and the bottom row is exact since preserves epimorphisms. ∎
We shall use the following lemma to identify with henceforth.
Lemma 4.3.9.
The canonical group homomorphism is an isomorphism.
The lemma would follow immediately from the discussion in Subsection 2.5 if were a local ring object, but this is not the case in general. Using Theorem 3.3.8, we shall see that functions as a “semilocal” ring object and therefore the isomorphism still holds.
Proof.
We need to show that for all , any -automorphism of becomes an inner automorphism after passing to a covering . Let . Then is the -fixed subring of . By Theorem 3.3.8, is semilocal. It is then well known that is an inner automorphism, [knus_quadratic_1991, III.§5.2]. Write for some . There exists such that is the image of an element in , also denoted , and such that agrees with on , e.g. if they agree on an -basis of . Since is a local ring object, and since , there exists a covering such that . By construction, factors through , and thus is inner on for all , as required. ∎
Lemma 4.3.10.
Let be a geometric morphism of topoi such that preserves epimorphisms. Then preserves quotients by equivalence relations. In particular, for any group object , any -torsor , and any -set , there is a canonical isomorphism .
Proof.
Recall that in a topos , any equivalence relation is effective, meaning that for some epimorphism —in fact, must be isomorphic to .
Since preserves epimorphisms and limits, this means that is canonically isomorphic to . ∎
We now come to the main result of this section, which allows passage from Azumaya algebras in the locally ringed topos to Azumaya algebras in the ringed topos .
Theorem 4.3.11.
Suppose is a locally ringed topos with involution , and that is an exact quotient relative to . Let and . Then the following categories are equivalent:
-
(a)
, the category of Azumaya -algebras of degree .
-
(b)
, the category of -torsors on .
-
(c)
, the category of Azumaya -algebras of degree .
-
(d)
, the category of -torsors on .
The equivalence between (a) and (b), resp. (c) and (d), is the one given in Proposition 2.5.2, and the equivalence between (a) and (c), resp. (b) and (d), is given by applying .
In the context of Example 4.3.3, where our exact quotient is induced from a good -quotient of schemes , the theorem says that every Azumaya algebra over admits an étale covering of (not of ) such that becomes a matrix algebra after base change to , and every automorphism of admits an étale covering of (again, not of ) such that becomes an inner automorphism after passing to . Theorem 4.3.11 can be regarded as a generalization of this fact.
Proof.
The equivalence between (a) and (b), resp. (c) and (d), is Proposition 2.5.2; here we identified with as in Lemma 4.3.9. The equivalence between (b) and (d) is Theorem 4.3.6(i), together with the isomorphism of Lemma 4.3.8. It now follows from Lemma 4.3.10 that the induced equivalence between (a) and (c) is given by applying . ∎
Remark 4.3.12.
The exact quotient gives rise to a morphism of ringed topoi with involution by setting to be the identity. The induced transfer functor is then an inverse to the equivalence .
Remark 4.3.13.
As phrased, Theorem 4.3.11 addresses Azumaya -algebras of constant degree only. These constitute all Azumaya algebras when is connected, but not in general. If we replace with a global section of the constant sheaf on , then Theorem 4.3.11 still holds, provided that is fixed by , in which case can be understood as an element of . Since Theorem 4.3.11 is used throughout, we tacitly assume henceforth that all Azumaya -algebras have degrees that are fixed under . This makes little difference in practice, because any Azumaya -algebra is Brauer equivalent to another Azumaya -algebra of degree which is fixed by .
We define an involution on the ringed topos by setting , and . Since preserves products, for any -algebra , we have as -algebras. However, we alert the reader that while as noncommutative rings, the -algebra structure of is obtained from the -algebra structure of by twisting via , as explained in Subsection 4.1. Theorem 4.3.11 now implies:
Corollary 4.3.14.
For all , the functor induces an equivalence between , the category of degree- Azumaya -algebras with -involution, and , the category of degree- Azumaya -algebras with -involution.
At this point we conclude that results proved so far allow one to shift freely between and , at least when Azumaya algebras, possibly with a -involution, are concerned. Of these two contexts, we shall work most often in the second, since this is technically easier. That said, the starting point is always a locally ringed topos with involution , and the choice of the corresponding , and is not in general uniquely determined by the initial data.
4.4. Examples of Exact Quotients
We now turn to providing various examples of exact -quotients. In particular, we will prove that Examples 4.3.3 and 4.3.4 are exact -quotients. It will also be shown that any locally ringed topos with involution admits an exact quotient.
Of the two conditions of Definition 4.3.1, condition (E2) is harder to establish. The following lemma is our main tool in proving it.
Lemma 4.4.1.
Let , be topoi and let be a geometric morphism. Suppose that has a conservative family of points with the property that for each , there exists a set of points such that the functors and from to are isomorphic. Then preserves epimorphisms.
Proof.
If is a point as in the lemma, then preserves epimorphisms because each does. By assumption, a morphism in is an epimorphism if and only if is an epimorphism for any as in the statement, so preserves epimorphisms. ∎
Informally, a geometric morphism satisfying the conditions of Lemma 4.4.1 can be regarded as having “discrete fibres”. It can also be thought of as a generalization of a finite morphism in algebraic geometry, thanks to the following corollary.
Corollary 4.4.2.
Let be a finite morphism of schemes. Then the direct image functor preserves epimorphisms.
This arises in the proof that the higher direct images vanish for cohomology with abelian coefficients, [de_jong_stacks_2017, Tag 03QN]. We have included a proof here in order to present a modification later.
Proof.
Recall ([artin_theorie_1972, Exp. VIII, §3–4]) that the points of are constructed as follows: Given , choose a cofiltered system of étale neighbourhoods such that , where is a strictly henselian ring, necessarily isomorphic to the strict henselization of . Then the functor from to and its right adjoint define a point , and these points form a conservative family, [artin_theorie_1972, Thm. VIII.3.5].
Let be such a point and write . For all sheaves on , we have . Note that . Since is finite, where is a finite -algebra.
By [de_jong_stacks_2017, Tag 04GH], where each is a henselian ring. Since the residue field of each is finite over the separably closed residue field of , each is strictly henselian. Letting be the primitive idempotents of , we may assume, by appropriately thinning the family , that are defined as compatible global sections on each . This allows us to write as such that for all . Let and denote the images of the closed point of in and , respectively, and let denote the point corresponding to the filtered system . Since we can commute a directed colimit past a finite limit, we have shown that
and the result now follows from Lemma 4.4.1. ∎
Corollary 4.4.3.
Let be a continuous morphism of topological spaces such that:
-
(1)
For any and any open neighbourhood , there exists an open neighbourhood of such that .
-
(2)
For any , the fibre is finite and, letting denote the points lying over , there exist disjoint open sets such that .
Then preserves epimorphisms.
The hypotheses are satisfied when is a finite covering space map of Hausdorff spaces, or a closed embedding of Hausdorff spaces, for instance.
It is also easy to see that condition (1) is equivalent to being closed, and condition (2) is equivalent to having finite fibres and being separated in the sense that the image of the diagonal map is closed.
Proof.
Again, we use Lemma 4.4.1. For , the points are induced by inclusion maps as rages over , [mac_lane_sheaves_1992, Chap. VII, §5]. Fix such an inclusion, let denote the points in , and let denote the inclusion maps. The corresponding morphisms on the topoi of sheaves will be denoted by the same letters.
Theorem 4.4.4.
Proof.
-
(i)
The fact that preserves epimorphisms is shown as in the proof of Corollary 4.4.2, except one has to replace [de_jong_stacks_2017, Tag 04GH] by Corollary 3.3.11. Checking that is the coequalizer of , amounts to showing that for any étale morphism , is the fixed ring of in . In fact, it is enough to check this after base changing to an open affine covering , so we may assume that factors as with open and affine. Write and . Since is affine, we may further write . The assumption that is a good quotient relative to implies that the sequence of -modules is exact. Since is flat over , the sequence is exact, and hence is the fixed ring of in .
-
(ii)
Conditions (1) and (2) of Corollary 4.4.3 are easily seen to hold, hence preserves epimorphisms. It remains to show that is the equalizer of under the action of . To this end, let be an open set. The -action on restricts to an action on and . In particular, is in natural bijection with the set of functions in that are fixed under the -action. This means that is the fixed subsheaf of under the action of .∎
Remark 4.4.5.
The proofs of Theorem 4.4.4(i) and Corollary 4.4.2 can be modified to work for the Nisnevich site of a scheme — simply replace étale neighbourhoods by Nisnevich neighbourhoods and strictly henselian rings by henselian rings. Disregarding set-theoretic problems, the large étale and Nisnevich sites can be handled similarly, using suitable conservative families of points, provided one assumes in Theorem 4.4.4(i) that splits in the middle, which is the case when . This assumption guarantees that the sequence remains exact after base-change to any -scheme , not necessarily flat.
The next examples bring several situations where condition (E1) of Definition 4.3.1 is satisfied while condition (E2) is not. They also show that some of the assumptions made in Theorem 4.4.4 cannot be removed in general.
Example 4.4.6.
Let be a strictly henselian discrete valuation ring with fraction field . Let denote the scheme obtained by gluing two copies of along , and let denote the involution exchanging these two copies. The morphism which restricts to the identity on each of the copies of is a geometric quotient relative to in the sense of [de_jong_stacks_2017, Tag 04AD], namely, , as topological spaces, and the latter property holds after base change. In particular, is the -quotient of in the category of schemes. However, the induced -equivariant morphism is not an exact quotient, the reason being that does not preserve epimorphisms.
To see this, fix a non-trivial abelian group , which will be regarded as a constant sheaf on the appropriate space, and let denote the inclusions of the two copies of in . Since is an open immersion, we can form the extension-by- functor , which is left adjoint to . Let . The counit maps and give rise to a morphism given by on sections. This morphism is surjective, as can be easily seen by checking the stalks. However is not surjective, as can be seen by noting that , , and any étale covering of has a section, because is strictly henselian.
Example 4.4.7.
Let be an infinite set endowed with the cofinite topology, let be an involution acting freely on , and let be the quotient map. Then the induced -equivariant morphism is not an exact -quotient, because fails to preserve epimorphisms. This is shown as in Example 4.4.6, except here one chooses and uses the open embeddings and . We conclude that in Theorem 4.4.4(ii), the assumption that is Hausdorff in cannot be removed in general, even when acts freely on .
Example 4.4.8.
Let be a principal ideal domain admitting exactly two maximal ideals, and . Suppose that there exists an involution exchanging and , and moreover, that the fixed ring of , denoted , is a discrete valuation ring. Let , and let be the morphism adjoint to the inclusion . Then is a good quotient relative to , but the induced -equivariant morphism is not an exact quotient, because, yet again, does not preserve epimorphisms. Again, this is checked as in Example 4.4.6 by using the open embeddings and . This shows that we cannot, in general, replace the étale site with the Zariski site in Theorem 4.4.4(i), even when is quadratic étale.
Remark 4.4.9.
Let be a scheme, let be an involution and let be a good quotient relative to . Then the associated morphism of topoi is not exact in general, even when is an fppf morphism.
For example, let be a field characteristic , and consider , and the -involution sending to . Then is surjective as morphism in , but its pushforward to is not, because is not in the image of for all commutative -algebras .
Nevertheless, when is finite and locally free, Theorem 4.3.11 and Corollary 4.3.14 still hold, the reason being that and are both represented by smooth affine group schemes over (use [bosch_1990_Neron_models, Prop. 7.6.5(h)]), and hence their étale and fppf cohomologies coincide [grothendieck_1968_groupe_de_Brauer_III, Thm. 11.7, Rmk. 11.8(3)]. As a result, some theorems in the next sections, e.g. Theorems 5.2.13 and 6.3.3, also hold in the context of ringed topoi associated to a finite locally free good -quotient of schemes .
We finish with demonstrating that every locally ringed topos with involution admits a canonical exact quotient, sometimes called the “homotopy fixed points”, as in [merling_equivariant_2017, Section 2]. We denote this exact quotient by .
As the notation suggests, when for a scheme , the ringed topos will be equivalent to the étale ringed topos of the Deligne–Mumford stack . Indeed, the objects of will be -equivariant sheaves, the data of which are equivalent to specifying a sheaf on the étale site of ; this is explained for coherent sheaves in [vistoli_1989_intersection_theory, Example 7.21], but the principle works for set-valued sheaves (in the sense of [de_jong_stacks_2017, Tag 06TN]) as well.
Construction 4.4.10.
Define the category as follows: The objects of consist of pairs , where is an object of and is a morphism satisfying . In other words, the objects of are objects of equipped with an involution, or a -action. Morphisms in are defined as commuting squares
Define to be the forgetful functor , and define to be the functor sending to where is the interchange morphism. For a morphism in , let . The functor is easily seen to be left adjoint to with the unit and counit of the adjunction given by and on the level of sections (in ).
For objects in and in , let denote the interchange morphism and let denote . Then is a natural isomorphism and is a natural isomorphism . We alert the reader that these natural isomorphisms are in general not the identity transformations, even when the involution is trivial.
Define the ring object in to be with the obvious ring structure. Finally, define to be , where the underlying morphism is given by on sections.
Proposition 4.4.11.
In Construction 4.4.10, the following hold:
-
(i)
is a Grothendieck topos.
-
(ii)
is an essential geometric morphism of topoi.
-
(iii)
A family of morphisms in is a covering if and only if is a covering in .
-
(iv)
is a local ring object in .
-
(v)
defines an exact quotient relative to .
Proof.
We first introduce the functor given by sending an object to where is the interchange morphism. It is routine to check that is left adjoint to ; the unit map is the canonical embedding and the counit map is the map restricting to on and to on . The existence of adjoints implies formally that is continuous and cocontinuous, and that and preserve epimorphisms. We now turn to the proof itself.
-
(i)
We verify Giruad’s axioms for . Briefly, if is a set of generators for , then is a set of generators for . Indeed, let be distinct morphisms in . Since is faithful, are distinct in , hence there exist and such that . By the adjunction between and , the morphism corresponds to a morphism in such that , as required.
That sums are disjoint and equivalence relations are effective in can be checked with the help of the forgetful functor and the fact that these properties hold in . Finally, the existence of colimits and the fact that they commute with fiber products can be checked directly.
-
(ii)
This is immediate from the adjunctions between , and noted above.
-
(iii)
We may replace with their disjoint union, denoted , to assume that consists of a single element. We need to show that is an epimorphism in if and only if is an epimorphism in . The “if” part follows from the fact that preserves epimorphisms, being a left adjoint. To see the converse, it is enough to show that the composition is an epimorphism. Let . Then , and under this isomorphism the map
is the composition , which is injective since an epimorphism. Thus, is an epimorphism.
-
(iv)
Let be -sections of generating the unit ideal. Then generate the unit ideal in , and hence there exists a covering such that for all . (We remark that is the -ring, in which the unique element is invertible; it is possible that some of the called for in this covering are initial objects.)
Fix . The adjunction between and gives rise to a morphism , adjoint to , and an isomorphism of rings . Unfolding the definitions, one finds that under this isomorphism, corresponds to , which is invertible. Thus, is invertible in . By (iii), the collection is a covering, so we have shown that is locally ringed by .
-
(v)
One checks that is the morphism
and so is the equalizer of . That preserves epimorphisms can be checked directly using the definitions and (iii). The verification of the coherence conditions in Definition 4.2.1 is routine. ∎
4.5. Ramification
Let be a locally ringed topos with an involution , and let be an exact quotient, see Definition 4.3.1. For brevity, write
As in Subsection 4.3, we write as . The automorphism is an involution the fixed ring of which is .
Definition 4.5.1.
Let be an object of . We say that is unramified (relative to ) along if is a quadratic étale -algebra in , see Subsection 3.2. Otherwise, is said to be ramified along .
The morphism is said to be unramified if it is unramified along , and ramified otherwise. It is everywhere ramified if is ramified along every non-initial object of .
We alert the reader that, contrary to the common use of the term “ramification”, we consider trivial -quotients as everywhere ramified.
Example 4.5.2.
Suppose is given a weakly trivial involution and is the trivial -quotient, namely, the identity map (Example 4.3.5). Then is everywhere ramified. Indeed, in this case and and . Since is a local ring object, for any in , the ring is nonzero, and so cannot be locally free of rank over .
For any object of , define to be a singleton if is unramified along , and an empty set otherwise. It follows from Lemma 3.2.2 that defines a sheaf (the action of on morphisms in is uniquely determined), which is then represented by an object of , denoted . We call the unbranched locus of . It is a subobject of . Clearly, is unramified if and only if , and is everywhere ramified if and only if .
The following propositions give a more concrete description of the unbranched locus when is induced by a -quotient of schemes or topological spaces, see Examples 4.3.3 and 4.3.4.
Proposition 4.5.3.
In the situation of Example 4.3.3, i.e., when is obtained from a good -quotient of schemes by taking étale ringed topoi, the unbranched locus of , defined above, is represented by an open subscheme . The latter can be defined in any of the following equivalent ways:
-
(a)
is the largest open subset of such that is quadratic étale.
-
(b)
The (Zariski) points of are those points such that is quadratic étale.
-
(c)
The (Zariski) points of are those points such that is quadratic étale; here, is the strict henselization of .
-
(d)
The (Zariski) points of are those points such that the set is a singleton , and induces the identity on .
Consequently, is unramified if and only if is quadratic étale.
Proof.
By virtue of Lemma 3.2.2, if is an étale morphism having image in , then is quadratic étale if and only if is quadratic étale. It follows that there exists a maximal open subset of with the property that is quadratic étale, and any as above factors through the inclusion . We also let denote the sheaf it represents in .
Since is a subobject of , the set is a singleton or empty for all . To show that is the unbranched locus of , it is enough to show that is unramified along an object of if and only if there exists a morphism . For any such , we can find a covering with each represented by some in . By Lemma 3.2.2, unramified along if and only if is unramified along each . Furthermore, if for each there is a morphism (in ), then these morphisms must patch to a morphism , because is a singleton. It is therefore enough to show that is unramified along if and only if there exists a morphism in . The latter holds precisely when , and so the claim follows from the definition of .
We finish by showing that the different characterizations of are equivalent. The equivalence of (a) and (b) follows from Corollary 3.3.5, and the equivalence of (b) and (d) follows from Theorem 3.3.8. That the condition in (b) implies the condition in (c) is clear. It remains to prove the converse. Since is faithfully flat over , this is a consequence of Lemma 3.2.2 applied to the fpqc site of . ∎
Proposition 4.5.4.
In the situation of Example 4.3.4, i.e., when is induced by a -quotient of Hausdorff topological spaces , the unbranched locus is represented by an open subset . Specifically, . Consequently, is unramified if and only if acts freely on .
Proof.
This is similar to the previous proof and is left to the reader. ∎
We refer to the situations of Examples 4.3.3 and 4.3.4 as the scheme-theoretic case and topological case, respectively. In both cases, we define the branch locus of to be the complement , where is as in Proposition 4.5.3 or Proposition 4.5.4. The ramification locus of is . In the scheme-theoretic case, we also endow and with the reduced induced closed subscheme structure.
Notice that in the topological case, is a double covering, while is a homeomorphism. With slight modification, a similar statement holds for schemes.
Proposition 4.5.5.
In the notation of Proposition 4.5.3, let , and regard and as reduced closed subschemes of and , respectively. Then:
-
(i)
is quadratic étale.
-
(ii)
restricts to the identity morphism on .
-
(iii)
is a homeomorphism, and when is invertible on , it is an isomorphism of schemes.
Proof.
- (i)
-
(ii)
Condition (d) of Proposition 4.5.3 implies that fixes the (Zariski) points of . Let be a (Zariski) section of and let be the largest open subset on which is invertible. Let . Then is invertible in , which is impossible by condition (d) of Proposition 4.5.3. Thus, . Since is reduced, we conclude that .
-
(iii)
It is enough to prove the claim after restricting to an open affine covering of , so assume , , with a radical ideal of , and , where denotes the radical of . The morphism is adjoint to the evident homomorphism , and induces an involution having fixed ring .
We know that is continuous and it is a set bijection since its set-theoretic fibers consist of singletons by condition (d) of Proposition 4.5.3. Thus, proving that is a homeomorphism amounts to checking that it is closed. Since any satisfies , the morphism is integral, and therefore closed by [de_jong_stacks_2017, Tag 01WM].
Suppose now that . We need to show that is bijective. Let . By virtue of (ii), induces the identity involution on , and thus . Since , we have established the surjectivity of . Next, write . Since is bijective, is contained in every prime ideal of , and since is reduced, . ∎
Remark 4.5.6.
-
(i)
In Proposition 4.5.5(iii), it is in general necessary to assume that is invertible in order to conclude that is an isomorphism. Consider, for example, a DVR with having a non-perfect residue field of characteristic , let be an element such that its image in is not a square, let , and let be the -involution sending to . Taking and , the set consists of the closed point of , but the induced map not an isomorphism.
-
(ii)
Let be a good -quotient of schemes which is everywhere ramified, and suppose is invertible on . Then Proposition 4.5.5 implies that the induced morphism is an isomorphism. However, in general, and in contrast to Proposition 4.5.4, it can happen that is not an isomorphism. For example, take and let be the -involution taking to .
Remark 4.5.7.
For a general exact -quotient with unbranched locus , it is possible to define the “branch topos” of as the full subcategory of consisting of objects such that the projection is an isomorphism. In the situation of Examples 4.3.3 and 4.3.4, this turns out to give the topos of sheaves over the set-theoretic or scheme-theoretic branch locus of defined above. We omit the details as they will not be needed in this work.
We finish with showing that the exact quotient of Construction 4.4.10 is unramified. Thus, any locally ringed topos with involution admits an unramified exact quotient. When is the étale ringed topos of a scheme , this generalizes the well-known fact that the morphism from to its quotient stack is quadratic étale.
Proposition 4.5.8.
The exact quotient of Construction 4.4.10 is unramified.
Proof.
Recall from Construction 4.4.10 that and , where is the interchange involution, and the morphism is given by on sections (in ). We shall make use of , the left adjoint of constructed in the proof of Proposition 4.4.11.
Write and observe that the unique map is a covering by Proposition 4.4.11(iii). By Lemma 3.2.2, it is enough to show that is a quadratic étale -algebra. In fact, we will show that .
We first observe that the slice category is equivalent to ; the equivalence is given by mapping in to , and by in the other direction. Now, consider and as sheaves on . Then and are sheaves of rings on the equivalent topos . Since is left adjoint to , for all objects of , we have natural isomorphisms of rings
and under these isomorphisms, the embedding induced by is given by on sections. From this it follows that as -algebras, and hence as -algebras. ∎
5. Classifying Involutions into Types
The purpose of this section is to classify involutions of Azumaya algebras into types in such a way which generalizes the familiar classification of involutions of central simple algebras over fields as orthogonal, symplectic or unitary.
Throughout, denotes a locally ringed topos with an involution and is an exact quotient relative to , see Definition 4.3.1. For brevity, we write
and, abusing the notation, denote the involution by . Theorem 4.3.11 and Corollary 4.3.14 imply that Azumaya -algebras with a -involution are equivalent to Azumaya -algebras with a -involution, and the latter are easier to work with.
In fact, most of the results of this section can be phrased with no direct reference to or the quotient map , assuming only a locally ringed topos , an -algebra , and an involution having fixed ring .
We remind the reader that Azumaya -algebras are always assumed to have a degree which is fixed by , see Remark 4.3.13. This is automatic when is connected.
5.1. Types of Involutions
Suppose is a field and is an involution, the fixed subfield of which is . Classically, when , the -involutions of central simple -algebras are divided into two types, orthogonal and symplectic, whereas in the case , they are simply called unitary; see [knus_book_1998-1, §2]. This classification satisfies the following two properties:
-
(i)
If and are central simple -algebras with -involutions such that , then and are of the same type if and only if and become isomorphic as algebras with involution over an algebraic closure of .
-
(ii)
If is a central simple -algebra with -involution, then has the same type as given by .
Of these two properties, it is mainly the first that motivates the classification into types. The second property should not be disregarded as it guarantees, at least when , that involutions adjoint to symmetric bilinear forms (resp. alternating bilinear forms, hermitian forms) of arbitrary rank all have the same type, see [knus_book_1998-1, §4].
Our aim in this section is to partition the -involutions of Azumaya -algebras into equivalence classes, called types, so that properties analogous to (i) and (ii) hold. To this end, we simply take the minimal equivalence relation forced by the “if” part of condition (i) and condition (ii).
Definition 5.1.1.
Let , be two Azumaya -algebras with a -involution. Let denote the involution of . On sections, it is given by .
We say that and are of the same -type or type if there exist and a covering in such that
as -algebras with involution.
Being of the same -type is an equivalence relation. The equivalence classes will be called -types or just types, and the type of will be denoted
The set of all -types will be denoted . Tensor product of Azumaya algebras with involution endows with a monoid structure. We write the neutral element, represented by , as .
We shall shortly see that our definition gives the familiar types in the case of fields, as well as in a number of other cases. It is no longer clear whether two involutions of the same degree and type are locally isomorphic, however, and the majority of this section will be dedicated to showing that this is indeed the case under mild assumptions. Another drawback of the definition is that it not clear how to enumerate the types it yields, and nor does it provide a way to test whether two involutions are of the same type. These problems will also be addressed, especially in the situation of Theorem 4.4.4, namely, in cases induced by a good -quotient of schemes on which is invertible, or by a -quotient of Hausdorff topological spaces.
Remark 5.1.2.
Let , be two Azumaya -algebras in with a -involution. Using Corollary 4.3.14, we say that and have the same -type (relative to ) when the same holds after applying . The equivalence class of is denoted or and the monoid of types is denoted .
We warn the reader that the -type of a -involution of an Azumaya -algebra depends on the choice of the quotient , which is why we include in the notation.
For instance, we shall see below in Theorem 5.2.17 that in the case where is given the trivial involution and is connected, then under mild assumptions, taking to be the trivial exact quotient results in two -types, whereas taking to be the exact quotient of Construction 4.4.10 results in only one -type.
Example 5.1.3.
-
(i)
Let be a connected scheme on which is invertible, let be the trivial involution and let . Consider the exact quotient obtained from by taking étale ringed topoi, see Example 4.3.3. In this case, , and . Thus, an Azumaya -algebra with a -involution is simply an Azumaya -algebra with an involution of the first kind on . It is well known that there are two -types: orthogonal and symplectic. The orthogonal type is represented by and the symplectic type is represented by , where is given by on sections and . Moreover, every Azumaya -algebra of degree with an involution of the first kind is locally isomorphic either to or to , where in the latter case and is given sectionwise by with ; see [knus_quadratic_1991, III.§8.5] or [parimala_92, §1.1]. In this case, is isomorphic to the group .
-
(ii)
Let be a quadratic étale morphism, and let denote the canonical -involution of . Again, let denote the exact quotient obtained from by taking étale ringed topoi. In this case, is quadratic étale over , and -involutions are known as unitary involutions. There is only one type in this situation, and moreover, any Azumaya -algebra of degree with a -involution is locally isomorphic to ; these well-known facts can be found in [parimala_92, §1.2] without proof, but they follow from the results in the sequel. We were unable to find a source providing complete proofs.
Example 5.1.4.
Let be a perfect field of characteristic , and consider the case of the trivial involution on . As in Example 5.1.3(i), this corresponds to taking , and . Azumaya -algebras with a -involution are therefore central simple -algebras with an involution of the first kind. There are again two types in this case, again called orthogonal and symplectic, but is isomorphic to the multiplicative monoid with corresponding to the symplectic type; see [knus_book_1998-1, §2]. This shows that the theory in characteristic is substantially different from that in other characteristics.
The assumption that is perfect can be dropped if one replaces the étale site with the fppf site (consult Remark 4.4.9).
5.2. Coarse types
In Subsection 5.1, we defined the type of a -involution of an Azumaya -algebra in terms of the entire class of algebras and not as an intrinsic invariant of the involution. We now introduce another invariant of -involutions, called the coarse type, which, while in general coarser than the type, will enjoy an intrinsic definition. It will turn out that under mild assumptions the invariants are equivalent, and this will be used to address the questions raised in Subsection 5.1. Apart from that, coarse types will also be needed in proving the main results of Section 6.
We begin by defining the abelian group object of via the exact sequence
| (3) |
and the abelian group object of via the short exact sequence
| (4) |
The group should be regarded as the group of elements of -norm . We call the global sections of coarse -types and write
The following example and propositions give some hints about the structure of .
Example 5.2.1.
If the involution of is trivial and the quotient map is the identity, then and the map is trivial, hence and .
Proposition 5.2.2.
If is unramified along an object of , see Subsection 4.5, then in . In particular, when is unramified, and .
When has enough points, it is possible to argue at stalks, and therefore the proposition follows from our version of Hilbert’s Theorem 90, Proposition 3.1.4(iii). The following argument applies even without the assumption of enough points.
Proof.
We must show that for all objects of and all satisfying , there is a covering and such that in .
Refining if necessary, we may assume that is a quadratic étale -algebra, see Subsection 3.2. By Proposition 3.1.4(iii), for all , there is such that . For each , choose some such that is the image of an element in , also denoted , which satisfies in . The set is not contained in any proper ideal of and therefore generates the unit ideal. Since is a local ring object, there exists a covering such that the image of is invertible in for all . Now take and to be the image of in . ∎
Proposition 5.2.3.
is a -torsion abelian sheaf.
Proof.
Let be an object of and . By passing to a covering of , we may assume that is the image of some . Since , we have , and thus in . ∎
Let . Using Lemma 4.3.9, we shall freely identify the group with . We denote by
the automorphism of given by on sections. This automorphism induces an automorphism on , which is also denoted . We need the following lemma.
Lemma 5.2.4.
Let be a section of and view as an -algebra isomorphism . Then .
Proof.
Suppose for some object of . It is enough to prove the equality after passing to a covering . We may therefore assume that is inner, and the lemma follows by computation. ∎
Construction 5.2.5.
Let be a degree- Azumaya -algebra with a -involution. We now construct an element
and call it the coarse -type of . This construction, which is concluded in Definition 5.2.7, will play a major role in the sequel.
Choose a covering such that there exists an isomorphism of -algebras . The isomorphism gives rise to a -involution
From and the involution , we construct
By Lemma 5.2.4, we have , hence . Replacing by a covering if necessary, we may assume that lifts to a section
From , we get
| (5) |
Note that , hence . Let be the image of in .
Lemma 5.2.6.
The section determines a global section of . It is independent of the choices made in Construction 5.2.5.
Proof.
Let denote the Čech hypercovering associated to —for the definition see Example 2.3.1. In particular, , and are given by on sections. Proving that determines a global section of amounts to showing that there exists a covering and such that holds in .
For , let denote the pullback of along . Define similarly. Let
and regard as an element of . The fact that for implies that . Therefore, using Lemma 5.2.4, we get
or equivalently, in .
There exists a covering such that the image of in , lifts to
The relation now implies that
| (6) |
in . This establishes the first part of the lemma.
Let denote the global section determined by . The construction of involves choosing , and above. Suppose that was obtained by replacing these choices with , and . We need to show that .
Define as above using , , in place of , , . It is clear that refining the covering does not affect . Therefore, refining and to , we may assume that . Write with . Then , and using Lemma 5.2.4, we get . Refining further, if necessary, we may assume that lifts to . The relation implies that there is such that . Thus,
and in . This completes the proof. ∎
Definition 5.2.7.
Let be an Azumaya -algebra with a -involution. The coarse -type or coarse type of is the global section of determined by constructed above. It shall be denoted or .
Remark 5.2.8.
In accordance with Remark 5.1.2, we shall write the coarse type of a -involution of an Azumaya -algebra, defined to be , as .
Proposition 5.2.9.
Let , be Azumaya -algebras with -involutions. Then:
-
(i)
for all .
-
(ii)
in .
-
(iii)
If there is a covering such that , then .
Consequently, the map is a well-defined morphism of monoids.
Proof.
Write and . Define as in Construction 5.2.5, and analogously, define using in place of .
-
(i)
The isomorphism gives rise to an isomorphism
Let , let and let . Straightforward computation shows that the image of in is . This means that coincides with in , and thus .
-
(ii)
Consider , let , and let . Define . Then maps onto , and we have , which means .
-
(iii)
Fix an isomorphism and, in the construction of , choose a covering factoring through . Taking and in the construction of , we find that
so . ∎
Remark 5.2.10.
Definition 5.2.11.
An abelian group object of is said to have square roots locally if the the squaring map is an epimorphism. That is, for any object of and , there exists a covering and such that .
Example 5.2.12.
If is the topos of a topological space with the ring sheaf of continuous functions into or the étale ringed topos of a scheme on which is invertible, then the group has square roots locally. Indeed, this holds at the stalks, because the stalks of are strictly henselian rings in which is invertible—this is well known in the case of an étale ringed topos of a scheme, or proved in Appendix A in the case of a topological space. Furthermore, if is the fppf ringed topos of an arbitrary scheme , then has square roots locally, because any -section has a square root over a degree- finite flat covering of .
The main result of this section is the following theorem, which shows that under mild assumptions, -involutions of Azumaya algebras of the same degree having the same coarse type are locally isomorphic. As a consequence, an analogue of the desired property (i) from Section 5.1 holds under the same assumptions. The theorem holds in particular when is induced by a good -quotient of schemes on which is invertible (see Example 4.3.3), or by a -quotient of Hausdorff topological spaces (see Example 4.3.4).
Theorem 5.2.13.
Let be a locally ringed topos with involution , let be an exact quotient relative to , and write and . Suppose that has square roots locally and at least one of the following conditions holds:
-
(1)
.
-
(2)
is unramified.
-
(3)
is odd.
Suppose and are two degree- Azumaya -algebras with -involutions. Then the following are equivalent:
-
(a)
and are locally isomorphic as -algebras with involution.
-
(b)
and have the same type.
-
(c)
and have the same coarse type.
Proof.
Corollary 5.2.14.
Suppose the assumptions of Theorem 5.2.13 hold and . Then the map is injective.
Proof.
Corollary 5.2.15.
With the assumptions of Corollary 5.2.14, is a -torsion group.
Remark 5.2.16.
The following theorem shows that the properties exhibited in Example 5.1.3 extend to our general setting under some assumptions.
Theorem 5.2.17.
Proof.
We note that part (i) applies in the case where is induced by a scheme on which is invertible endowed with the trivial involution ; see Example 5.1.3(i).
Part (ii) applies to the case where is induced by a quadratic étale morphism of schemes where is given the canonical -involution; see Example 5.1.3(ii).
Our last application of Theorem 5.2.13 provides a concrete realization of the first cohomology set of the projective unitary group of an Azumaya -algebra with a -involution . As usual, the unitary group of is the group object in whose -sections are , and the projective unitary group of is the quotient
where , the group of -norm elements in defined above.
If is another Azumaya -algebra with a -involution such that , we further define to be the sheaf of -linear isomorphisms from to , and to be the group sheaf of -linear automorphisms of .
Lemma 5.2.18.
Suppose has square roots locally. Let be a degree- Azumaya -algebra with a -involution. Then the map sending a section to conjugation by is an epimorphism with kernel . Consequently, it induces an isomorphism .
Proof.
That the kernel is follows easily from the fact that the centre of is . We turn to proving that the map is an epimorphism.
Let and . By replacing with suitable covering, we may assume that and that is given section-wise by for some . Since , for any section , we have , and thus . In fact, since , we have . By assumption, we can replace with a suitable covering to assume that there is with . Replacing with yields . We have therefore shown that over a covering of , lifts to a section of . ∎
Corollary 5.2.19.
With the assumptions of Theorem 5.2.13, let be a degree- Azumaya -algebra with a -involution and identify with as in Lemma 5.2.18. Then the functor defines an equivalence between the full subcategory of consisting of -algebras with a -involution of the same type as and . Consequently, is in canonical bijection with isomorphism classes of the aforementioned algebras with involution.
Proof.
Many of the previous results require that . Indeed, our arguments build on using the coarse type, which is too coarse if is not invertible—see Remark 5.2.10. Nevertheless, we ask:
Question 5.2.20.
A particularly interesting case is the morphism of fppf ringed topoi associated to a finite locally free good -quotient , where is a scheme on which is not invertible (consult Remark 4.4.9).
5.3. Proof of Theorem 5.2.13
In this subsection we complete the proof of Theorem 5.2.13 by showing that condition (c) implies condition (a). This result is given as Proposition 5.3.8 below. The reader can skip this subsection without loss of continuity.
In the following lemmas, unless otherwise specified, will be a ring, an involution, and will be the fixed ring of . We will write for , and for the involution induced on by . Let and let be an -hermitian matrix, which is to say
Let be the -hermitian form associated to ; it is given by where are written as column vectors. Let denote the reduction of to .
Lemma 5.3.1.
Assume is local. If , or in , or is odd, then there exists such that is a diagonal matrix.
Proof.
Proving the lemma is equivalent to showing that is diagonalizable, i.e., has an orthogonal basis.
We first claim that has an orthogonal basis. This is well known when is a field; see [scharlau_quadratic_1985, Th. 7.6.3] for the case where or in , and [albert_symmetric_1938, Thm. 6] for the case where , in and is odd. We note in passing that the second case can occur only when the characteristic of is . If is not a field, then Theorem 3.3.8 and Proposition 3.1.4(ii) imply that , where is the residue field of , and acts by interchanging the two copies of . In this case is hyperbolic and the easy proof is left to the reader.
We now claim that any nondegenerate -hermitian form whose reduction admits a diagonalization is diagonalizable, thus proving the lemma. Let be an orthogonal basis for and let be an arbitrary lift of . Then and hence . Write and . The -module is free because is semilocal and is projective of constant -rank . Furthermore, since , the form is diagonalizable by construction. We finish by applying induction to . ∎
Lemma 5.3.2.
Assume is local, and suppose , that and that . Then there exists such that is a direct sum of matrices in . In particular, is even.
Proof.
We need to show that admits a basis such that is orthogonal to whenever and such that .
By Theorem 3.3.8, the assumption implies that is local, hence is a field of characteristic different from and is a nondegenerate alternating bilinear form. This means that must be even. Choose a nonzero . Since is nondegenerate, there is such that . Since is alternating, we also have and . Let be lifts of and . The previous equations imply that . In particular, is invertible, and so . We proceed by induction on the restriction of to . ∎
Lemma 5.3.3.
Assume is local and . Then .
Proof.
Write and let . Then . The equality implies . Likewise, . We finish by noting that . ∎
In the following lemmas, given a ring and , we write to denote the ring and let denote the image of in . By induction, we define . If is an involution with fixed ring and , then extends to by setting , and the fixed ring of is .
Lemma 5.3.4.
Assume is local and suppose , that and . Suppose that lies in . Then there are , and () such that and in for all .
Proof.
Again, by Theorem 3.3.8, is local. We write , and let denote the standard -basis of . Once and above have been chosen, we need to show that for all , there are such that .
Our assumption on implies that . Replacing with , we may assume that and so . We further write and . Since , we have and , and since , it follows that and hence .
Observe that the polynomial has discriminant
and that . The roots of in are and and we note that
| (7) |
We repeat this construction with in place of , denoting the elements corresponding to , , by , , .
Fix a maximal ideal and write , , and .
It is clear that is local. We claim that is also local and . Indeed, by [reiner_maximal_1975-1, Th. 6.15], we have , and hence , which in turn implies . By Lemma 3.3.10, we have , and by Lemma 5.3.3, . Using the last three inclusions, we get , and since is semiprime, . The latter implies that factors through , and hence the specialization of to is the identity. Since is flat over , the local ring is the fixed ring of and Theorem 3.3.8 implies that is local.
Now, the inclusion implies . By equation (7), there is such that , and hence . In the same way, there is such that .
Working in , we have
| (8) | ||||
and likewise for . Write . Since , we have
| (9) |
and since , this implies . We further have by (8). In the same way, writing , we have , and . Let . Then, using (9), we get
Likewise, satisfies . Since , the vectors span . Since is nondegenerate, this forces . Replacing with , we find that .
Finally, for every maximal ideal , choose such that the coefficients of , constructed above are defined in and such that the identity holds in . Then . Since is a finite algebra over a local ring, it has only finitely many maximal ideals. The result follows. ∎
Lemma 5.3.5.
Maintaining the assumptions made at the beginning of this subsection, suppose that is another -hermitian matrix. Suppose further we are given a prime ideal , units , elements and () such that generate the unit ideal in and such that for all . Then there is for which the previous condition holds upon replacing , with , .
Proof.
There is such that are in the image of . We may replace , with , and assume henceforth.
Write and , and choose such that . Then there is such that are images of elements in , also denoted , and such that in . Again, we replace , with , and assume .
Fix . There are , and such that in . Since , we have in , and hence there is such that in . This in turn implies that there is such that in . Replacing , with , , we may assume . Let . Then the image of in is and the equality implies that in . Taking determinants, we see that .
The lemma follows by applying the previous paragraph to all . ∎
We now return to the context of ringed topoi.
Lemma 5.3.6.
Assume , and let , and . Then:
-
(i)
There is a covering of such that and .
-
(ii)
There exists a covering of such that and . Equivalently, there exists a covering of and () such that and .
Proof.
-
(i)
We note that this statement requires proof because is not a local ring object in general. Observe that and hence . Since and is a local ring object, the assumption implies that there exists a covering of such that and . It follows that and .
-
(ii)
Choose a covering as in (i) and let , . Since , we have , and hence . Likewise, . ∎
The following lemma is known when is unramified or trivial, i.e., when is a quadratic étale -algebra or . The ramified situation that we consider appears not to have been considered before in the literature.
Lemma 5.3.7.
Let , let , and let be two -hermitian matrices, i.e., and . Assume has square roots locally and that , or is unramified, or is odd. Then there exists a covering and such that in .
Proof.
Suppose first that . Let and be as in Lemma 5.3.6(ii). We may replace with and assume that henceforth.
We claim that it is enough to show that for all , there are , and () such that generate the unit ideal in and . If this holds, then Lemma 5.3.5 implies that for all , we can find such that the previous condition holds upon replacing with . Since and is a local ring object, there is a covering such that . Fix some and let be as above. By construction, the images of in are invertible in . Since has square roots locally, we can replace with a suitable covering such that have square roots in . In particular, factors through . Applying the fact that is a local ring object again, we see that there is a covering such that the image of in is invertible. It follows that factors through and hence there is — the image of — such that . Finally, let and take .
Let . We now prove the existence of above. Write and . Then is local and it is the fixed ring of . We write and let denote the involution induced by .
Suppose or . By Lemma 5.3.1, we may assume that and are diagonal, say and . Since and are -hermitian, we have , hence for all . Writing and , we have , as required (take ).
Suppose now that and . Applying Lemma 5.3.2 to and , we may assume that and are direct sums of matrices in , say , with . Applying Lemma 5.3.4 to , we obtain , , and such that .
Let , and regard as elements of . For every tuple , let , and , where are regarded as elements of . Then and , which is what we want. This establishes the lemma when .
To prove the remaining cases, we observe that when is unramified, or is odd, the use of Lemmas 5.3.2, 5.3.4 and 5.3.6 can be avoided, and hence the assumption is unnecessary.
We can finally complete the proof of Theorem 5.2.13.
Proposition 5.3.8.
Let be a positive integer. Suppose has square roots locally and at least one of the following holds:
-
(1)
.
-
(2)
is unramified.
-
(3)
is odd.
Let and be two degree- Azumaya -algebras with -involutions having the same coarse type. Then and are locally isomorphic as -algebras with involution.
Proof.
Following the construction of in 5.2, define , , , , and so that induces . Repeating the construction with in place of , we define analogously. By refining both and , we may assume .
Since , and determine the same section in . Thus, there exists a covering and such that . Replacing with , and with , we may assume that .
Now, by Lemma 5.3.7, there exists a covering and such that . Again, replace with . Letting denote the image of in , we deduce . Unfolding the construction in 5.2, one finds that , and likewise . Let . Then is an isomorphism of -algebras, and since , we have
Thus, defines an isomorphism of algebras with involution . ∎
5.4. Determining types in specific cases
Under mild assumptions, Theorem 5.2.13 provides a cohomological criterion to determine whether two -involutions of Azumaya algebras have the same type, and Corollary 5.2.14 embeds the possible -types in . We finish this section by making this criterion and the realization of the types even more explicit in case the exact quotient is induced by a -quotient of schemes or topological spaces.
Notation 5.4.1.
Throughout, we assume one of the following:
-
(1)
is a scheme on which is invertible, is an involution and is a good quotient relative to , see Example 4.3.3.
-
(2)
is a Hausdorff topological space, is a continuous involution, and is the quotient map.
We will usually treat both cases simultaneously, but when there is need to distinguish them, we shall address them as the scheme-theoretic case and the topological case, respectively.
In the scheme-theoretic case, the terms sheaf, cohomology and covering should be understood as étale sheaf, étale cohomology and étale covering, whereas in the topological case, they retain their ordinary meaning relative to the relevant topological space. Furthermore, in the topological case, stands for , the sheaf of continuous functions into , and likewise for all topological spaces.
As in Subsection 5.2, write and , and define to be the kernel of the -norm and to be the cokernel of . By means of Theorem 4.4.4, the results of the previous subsections can be applied, essentially verbatim, to Azumaya -algebras with -involution.
Recall from Propositions 4.5.3 and 4.5.4 that there is a maximal open subscheme, resp. subset, such that is unramified, i.e. a quadratic étale morphism or a double covering of topological spaces. We write
Then and are the branch locus and the ramification locus of , respectively. We endow and with the subspace topologies. In the scheme-theoretic case, we further endow them with the reduced induced closed subscheme structures in and , respectively. Recall from Proposition 4.5.5 and the preceding comment that induces an isomorphism of schemes, resp. topological spaces, , and restricts to the identity map on . In particular, .
Recall that denotes the sheaf of square roots of in ; we abbreviate this sheaf as . Since is invertible in , the sheaf is just the constant sheaf . Similar notation applies to .
Let be an Azumaya -algebra with -involution and let be a point of the the ramification locus. Propositions 4.5.3 and 4.5.4 imply that and the specialization of to , denoted , is the identity. Thus, the specialization of to , denoted , is a central simple -algebra with an involution of the first kind.
Lemma 5.4.2.
With the above notation, the function determined by
is locally constant, and therefore determines a global section .
Proof.
We may assume that is constant; otherwise, decompose into a disjoint union of components on which this holds and work component-wise.
Let denote the base change of from to , namely , where denotes the inclusion map. For any point , the type of at may be calculated relative to . We may therefore replace and with and to assume that and is the trivial involution.
Let and . That is, and are the sheaves and -symmetric and -antisymmetric elements in . Since , both and are -modules, and since is invertible on , the sequence is split exact. Consequently, the sequence remains exact after base changing to for all , and so we may identify with the -symmetric elements of .
It is well known [knus_book_1998-1, §2A] that equals when is orthogonal and when is symplectic. Since is an -summand of , it is locally free. Thus, the rank of is locally constant, and a fortiori so is . ∎
We will prove, after a number of lemmas, that the element determines the type of . In the course of the proof, we shall see that the sheaf introduced in Subsection 5.2 is nothing but the pushforward to of the sheaf on .
Lemma 5.4.3.
Consider a commutative diagram
in which is a good -quotient of schemes, resp. a -quotient of Hausdorff topological spaces, and is -equivariant. Let denote the involution of and let denote the sheaves corresponding to and constructed with in place of . Then:
-
(i)
There are commutative squares of ring sheaves on and , respectively:
Here, the horizontal arrows of the right square are the adjoints of the horizontal arrows of the left square relative to the adjuntion between and . Furthermore, in both squares, the top horizontal arrows are morphism of rings with involution.
-
(ii)
The left square of (i) induces morphisms , and . Furthermore, if is an Azumaya -algebra with a -involution and denotes the base change of to , namely , then the image of in is .
Proof.
Part (i) and the first sentence of (ii) are straightforward from the definitions. We turn to prove the last statement of (ii).
We first claim that
| (10) |
To see this, observe that the relevant counit maps induce a ring homomorphism
which respects the relevant involutions. This morphism is adjoint to a morphism
| (11) |
which we claim to be the desired isomorphism. This is easy to see when . In general, by Theorem 4.3.11, there exists a covering such that becomes a matrix algebra after pulling back to . Thus, is an isomorphism, and we conclude that so does (11).
With (10) at hand, let be as in Construction 5.2.5, applied to . We may assume that is represented by a covering of , denoted . Let be the pullback of along , which corresponds to the sheaf in . Let and let . The right square of (i) induces canonical maps , and (notice that is exact). Let be the image of in , and define and similarly. It is easy to check that we can apply Construction 5.2.5 to using . Consequently, the image of in agrees with the image of , which is exactly what we need to prove. ∎
Endowing with the trivial involution, we can apply Lemma 5.4.3 with the square
| (12) |
where is the restriction of to . By Example 5.2.1, the sheaf is just and hence Lemma 5.4.3(ii) gives rise to a morphism
Lemma 5.4.4.
is an isomorphism of abelian sheaves on .
Proof.
To show that is an isomorphism, it is enough to check the stalks. The topos-theoretic points of are recalled in the proofs of Corollaries 4.4.2 and 4.4.3; they are in correspondence with the set-theoretic points of .
Let be a point, corresponding to . Since is exact, is the kernel of and is the cokernel of .
Suppose that . Then, since is a closed embedding, . On the other hand, since is unramified at , it is unramified at a neighborhood of and hence by Proposition 5.2.2. Thus, is an isomorphism.
Suppose henceforth that . Then is ramified at . We claim that is local and induces the identity map on its residue field. This is evident from the definitions in the topological case, see Proposition 4.5.4. In the scheme-theoertic case, this follows from condition (c) in Proposition 4.5.3 and Theorem 3.3.8 after noting that .
Now, we have . With the notation of Lemma 5.4.3, applied to the square (12), the morphism is just a restriction of the morphism . This implies that the images of in are mapped under to , respectively, so is surjective.
To finish, we show that consists of at most elements. Every is represented by some . Since and is local, either or is invertible. Suppose . Since , we have , or rather, , which implies . Similarly, when , we find that . It follows that and the proof is complete. ∎
We finally prove the main result of this subsection.
Theorem 5.4.5.
Remark 5.4.6.
We do not know whether every arises as for some , see Remark 5.2.16.
Proof.
By Theorem 5.2.13 and Example 5.2.12, in order prove (i), it is enough to prove that , and this follows if we prove (ii).
Apply Lemma 5.4.3 and its notation to the square (12). The lemma gives rise to a morphism of sheaves , which is an isomorphism by Lemma 5.4.4. This in turn induces an isomorphism
such that is mapped to , where denotes the base change of to . Since is an isomorphism, this gives rise to an isomorphism
which we take to be . It remains to show that .
Since , and since the image of in is , it is enough to show that the image of in is . To this end, we replace and with and . Now, is the trivial involution and we may assume that and is the identity map. The map is just the identity map , see Example 5.2.1, and the proof reduces to showing that .
Let , and let be as in Construction 5.2.5, applied to . We may assume that the sheaf in is represented by a covering . Since is the trivial involution, , and since is the constant sheaf on , there is a covering such that and .
There is and such that is the image of under . It is immediate from the definition of that . Let denote the residue field of . By construction, , where is given section-wise by and . Thus, is orthogonal when and symplectic when . The same applies to . Since , it follows that is orthogonal when and symplectic when . This means , so we are done. ∎
6. Brauer Classes Supporting an Involution
6.1. Introduction
Let be a field and let be an involution with fixed field . The central simple -algebras admitting a -involution were characterized by Albert, Riehm and Scharlau, see for instance [knus_book_1998-1, Thm. 3.1], who proved:
Theorem.
Let be a central simple -algebra. Then:
-
(i)
(Albert) When , admits a -involution if and only if in .
-
(ii)
(Albert–Riehm–Scharlau) When , admits a -involution if and only if in .
Here, is the corestriction algebra of , whose definition we recall below.
The Albert–Riehm–Scharlau Theorem does not, in general, hold if we replace with an arbitrary ring. However, in [saltman_azumaya_1978], Saltman showed that the Brauer classes admitting a representative with a -involution can still be characterized similarly.
Theorem (Saltman [saltman_azumaya_1978, Th. 3.1]).
Let be a ring, let be an involution and let be the fixed ring of . Let be an Azumaya -algebra. Then:
-
(i)
When , there exists such that admits a -involution if and only if in .
-
(ii)
When is quadratic étale over , there exists such that admits a -involution if and only if in .
A later proof by Knus, Parimala and Srinivas [knus_azumaya_1990, Thms. 4.1, 4.2] applies in the generality of schemes and also implies that the representative can be chosen such that .
In this section, we extend Saltman’s theorem to locally ringed topoi with involution. We note that our generalization implies in particular that Salman’s theorem applies to topological Azumaya algebras. Furthermore, while Saltman’s theorem assumes that , or is quadratic étale over the fixed ring of , our result will apply without any restriction on the involution. Finally, we also characterize the possible types, or more precisely, coarse types, of the involutions of the various representatives .
Notation 6.1.1.
Throughout this section, let be a locally ringed topos with ring object and involution , and let be an exact quotient relative to , see 4.3. Recall that such quotients arise, for instance, from -quotients of schemes or Hausdorff topological spaces as explained in Examples 4.3.3 and 4.3.4. In such cases, we shall work with the original schemes, resp. topological spaces, denoted and , rather than the associated ringed topoi.
As in Section 5, we write and .
We sometimes omit bases when evaluating cohomology; the base will always be clear from the context. If is an abelian group in , we shall freely identify , written , with , written , using Theorem 4.3.6.
6.2. The Cohomological Transfer Map
The corestriction map considered in the aforementioned theorems of Albert–Riehm–Scharlau and Saltman is a special case of the cohomological transfer map, which will feature in our generalization of Saltman’s theorem.
Definition 6.2.1.
The cohomological -transfer map is the composite of the isomorphism induced by , see Theorem 4.3.6, and the morphism induced by the -norm map . When no confusion can arise, we shall omit , simply writing for , and calling it the transfer map.
Example 6.2.2.
If the involution of is weakly trivial and is the trivial quotient, see Example 4.3.5, then the -norm is the squaring map , and so is multiplication by .
Example 6.2.3.
Let be a quadratic étale morphism of schemes, and let be the canonical -involution of , given section-wise by . We consider the exact quotient obtained from and by taking étale ringed topoi, see Example 4.3.3. In this case, the transfer map is, by definition, the corestriction map . Moreover, restricts to a map which can be described explicitly on the level of Azumaya algebras: Let be an Azumaya -algebra. The corestriction algebra is an Azumaya -algebra defined as the -subalgebra of fixed by the exchange automorphism, given by on sections. The map is then given by , see [knus_azumaya_1990, p. 68] (the diagram on that page contains a misprint, on the right column, both ‘’s should be ‘’s).
Remark 6.2.4.
In contrast to the situation in Examples 6.2.2 and 6.2.3, we do not know whether
restricts to a map between the Brauer groups , even in the cases induced by a good -quotient of schemes . Some positive results appear in [auel_parimala_suresh_2015, Lem. 5.1, Rmk. 5.2]. Also, when is locally free of rank over , Ferrand [ferrand_1998_norm_functors] constructs a universal norm functor taking -algebras to -algebras, which coincides with when is quadratic étale over , but it is a priori not clear whether it takes Azumaya -algebras to Azumaya -algebras in general. We hope to address this problem in a subsequent work.
We further note that without assuming that is unramified, the construction of Example 6.2.3 may produce an algebra which is not Azumaya. For example, it can be checked directly that is not Azumaya over when , , and is the -involution taking to .
Example 6.2.5.
In the case where is a Hausdorff topological space with a free -action and is the corresponding -sheeted covering, the construction
is a special case of the usual transfer map for a -sheeted cover. This can be proved by considering on the level of -cocycles. See also [piacenza_transfer_1984, Sec. 3.3] and note that takes , the sheaf of nonvanishing continuous complex-valued functions on , to on .
Remark 6.2.6.
There is a notion of transfer for ramified covers where is a finite group, in particular, when . This may be found in [aguilar_transfer_2010]. It seems likely, that map given here is a special case of that construction, but we do not pursue this further.
6.3. Brauer Classes Supporting a -Involution
In this subsection, we characterize those Brauer classes in admitting a representative with a -involution, thus generalizing Saltman’s theorem [saltman_azumaya_1978, Th. 3.1].
We remind the reader that the notational conventions of Notation 6.1.1 are still in effect. In particular, is a local ring object in and is a commutative -algebra with involution such that the fixed ring of is .
As in Subsection 5.2, we define to be the kernel of the -norm and let be the quotient of by the image of the map . Recall that is the group of coarse -types and there is a map associating an Azumaya -algebra with a -involution to its coarse type, see Subsection 5.2.
The short exact sequence induces the connecting homomorphism
and the short exact sequence induces a connecting homomorphism
Notation 6.3.1.
We denote the composite morphism by ,
Proposition 6.3.2.
The map is the -map in the following cases:
-
(i)
When is a trivial quotient (Example 4.3.5), i.e. .
-
(ii)
When is everywhere ramified (Definition 4.5.1), and has square roots locally.
-
(iii)
When is unramified (Definition 4.5.1), i.e. is a quadratic étale -algebra.
-
(iv)
When is a good -quotient of schemes, is noetherian and regular, and is unramified at the generic points of ; the corresponding exact quotient is obtained by taking étale ringed topoi as in Example 4.3.3.
Proof.
-
(i)
In this case, is trivial. Since factors through , the result follows.
-
(ii)
We claim that squaring induces an automorphism of , and hence of the group . Since is a -torsion group (Proposition 5.2.3), this forces to vanish, implying vanishes as well.
We show the surjectivity of by checking that has square roots locally. Let be an object of and . Since has square roots locally, there a covering and such that . Replacing with and with , we may assume . Now, by Lemma 5.3.6, there is a covering and such that in and in . We may refine to assume that there is such that and get . Similarly, we refine to find a square root of in and conclude that has a square root on .
Next, let denote the kernel of . A section of is represented by some such that , or rather, . Since and , and since is a local ring object, there is a covering such that and . By virtue of Lemma 3.3.3, is a quadratic étale over , so our assumption that is everywhere ramified forces . Thus, is a covering, implying that is invertible in . Since , we must have , so . It follows that represents the -section in , and thus .
-
(iii)
In this case, a version of Hilbert’s Theorem 90 applies in the form of Proposition 5.2.2, and . A fortiori, is .
-
(iv)
We may assume that is connected and therefore integral, otherwise we may work component by component.
Let denote the generic point of . Since is flat, is a good -quotient relative to the action induced by ; denote the sheaves corresponding to by . We now apply Lemma 5.4.3 to the square
which gives rise to maps , , adjoint to the maps in the lemma. The exactness of together with the natural homomorphism now give rise to a commutative diagram
By [grothendieck_groupe_1968, Cor. 1.8], the right vertical morphism is injective (here we need to be regular), and by (iii), . Therefore, . ∎
We are now ready to state our generalization of Saltman’s theorem. Whereas Saltman’s original proof [saltman_azumaya_1978] and the later proof by Knus, Parimala and Srinivas [knus_azumaya_1990] make use of the corestriction of an Azumaya algebra, we cannot employ this construction, as demonstrated in Remark 6.2.4. Rather, our proof is purely cohomological, phrased in the language set in Subsections 2.3 and 2.4. We remind the reader of our standing assumption from Remark 4.3.13 that the degrees of all Azumaya -algebras considered are fixed under , which is automatic when is connected.
Theorem 6.3.3.
Let be a locally ringed topos with involution , let be an exact quotient relative to , and consider the map of Definition 6.2.1 and the map of Notation 6.3.1. Let be an Azumaya -algebra of degree , and let . Then there exists admitting a -involution of coarse type if and only if in . The algebra can be chosen such that .
We recover Saltman’s original theorem [saltman_azumaya_1978, Th. 3.1] and the improvement of Knus, Parimala and Sinivas [knus_azumaya_1990, Thms. 4.1, 4.2] from Theorem 6.3.3 by taking to be the exact quotient associated to a good -quotient of schemes such that is an isomorphism or quadratic étale, see Examples 4.3.3. In this case, by Proposition 6.3.2, and the transfer map coincides with multiplication by when , or with the corestriction map when is quadratic étale, as demonstrated in Examples 6.2.2 and 6.2.3.
The relation between the type and the coarse type of an involution, as well as the question of when two involutions of the same type are locally isomorphic, had been studied extensively in Subsections 5.2 and 5.4.
Proof.
Thanks to Theorems 4.3.6 and 4.3.11, we may replace with and work with -algebras, rather than -algebras. We abuse the notation and denote the map induced by as .
Suppose first that there exists admitting a -involution of coarse type . We may replace with . We now invoke all the notation of Construction 5.2.5 and the proof of Lemma 5.2.6 through which is constructed from . Specifically:
-
•
is a covering such that there exists an isomorphism of -algebras ,
-
•
is an involution of ,
-
•
is an element of ,
-
•
is a lift of (refine if necessary),
-
•
is an element of , embedded diagonally in ,
-
•
is the Čech hypercovering corresponding to , see Example 2.3.1,
-
•
, where is the pullback of along ,
-
•
is a lift of to , where is some covering,
-
•
is an element , embedded diagonally in ,
-
•
is the image of in ; it descends to a global section of since .
By Lemma 2.3.2, there is a hypercovering morphism such that factors through . We replace with .
Recall from Theorem 4.3.11 that corresponds to a -torsor, which in turn corresponds to a cohomology class in . We claim that is a -cocycle in which represents this cohomology class. Indeed, the -torsor corresponding to is , and by construction. By the isomorphism given in the proof of Proposition 2.4.2(i), the cohomology class corresponding to is represented by .
Consider the short exact sequence and its associated -term cohomology exact sequence, see Proposition 2.4.2(iii). It follows from the definition of , see the proof of Proposition 2.4.2(iii), that is represented by
| (13) |
and thus, is represented by .
On the other hand, by the definition of , see the beginning of this subsection and the end of 2.3, is represented by the image of in , since . Likewise, by the definition of , the class is represented by .
In order to show that , we check that in . For the computation, we shall make use of , , and the fact that if is central in a group , then .
This completes the proof of the “only if” statement.
Suppose now that . Define , , , , and as before. Using Lemma 2.3.2 twice, we can refine to assume that lifts to some and there is such that
| (14) |
in . As explained above, is represented by and is represented by . The assumption therefore means that, after refining , there exists such that
We replace with , which does not affect (14) and allows us to assume
| (15) |
Writing in block form, define the matrices
and let be the involution given by on sections. Also, let be the image of in , namely, is the automorphism of given by on sections.
We first observe that . Indeed, working in and using (13) and (15), we find that
| (16) |
Let denote the Čech hypercovering associated to , see Example 2.3.1. By Lemma 2.4.1, descends uniquely to a cocycle . The Čech -cocycle defines descent data for along , giving rise to an Azumaya -algebra of degree and an isomorphism such that , where is the pullback of along . Note that by construction, represents the class in corresponding to , hence (16) implies that .
We now claim that descends to an involution . Letting denote the pullback of along , and noting that is a covering, see Subsection 2.3, this amounts to showing that . To see this, we first note that (14) and imply that
or equivalently,
Using this, for any section of , we have
which is what we want.
We finish by checking that . To see this, apply Construction 5.2.5 to using , , and defined above and note that . ∎
We now specialize Theorem 6.3.3 to Azumaya algebras over schemes and over topological spaces.
It is worth recalling at this point that in the situation of a good -quotients of schemes such that is invertible on (Example 4.3.3), or a -quotient of Hausdorff topological spaces (Examples 4.3.4), the sheaf is isomorphic to , where is the embedding of the branch locus of in . Under this isomorphism, the coarse type of an involution is the unique global section such that if is orthogonal, and if is symplectic, for all ; see Subsection 5.4. Furthermore, in these situations, two -involutions of the same coarse type have the same type, and they are locally isomorphic if the degrees of their underlying Azumaya algebras agree; this follows from Theorem 5.2.13 and Corollary 5.2.14.
Corollary 6.3.4.
Let be a scheme, let be an involution and let be a good quotient relative to . Let be an Azumaya -algebra and let be a coarse type. Then there exists admitting a -involution of coarse type if and only if in . The algebra can be chosen such that .
Corollary 6.3.5.
In the situation of Corollary 6.3.4, suppose that
-
(1)
, or
-
(2)
is quadratic étale, or
-
(3)
is noetherian and regular, and is unramified at the generic points of .
Then there exists admitting a -involution if and only if . In this case, can be chosen to have a -involution of any prescribed coarse type (or any prescribed type, when is invertible on ) and to satisfy .
Corollary 6.3.6.
Let be a Hausdorff topological space, let be a continuous involution, and let denote the quotient map . Let be an Azumaya -algebra and let be a coarse type. Then there exists admitting a -involution of coarse type if and only if in . The algebra can be chosen such that .
Corollary 6.3.7.
In the situation of Corollary 6.3.6, if , or acts freely on , then there exists admitting a -involution if and only if . In this case can be chosen to have a -involution of any prescribed type and to satisfy .
Remark 6.3.8.
Let be a connected semilocal ring, and let be an involution with fixed ring . When or is quadratic étale over , it was observed by Saltman [saltman_azumaya_1978, Th. 4.4] that an Azumaya -algebra that is Brauer equivalent to an algebra with a -involution already possesses a -involution. Otherwise said, in this special situation, we can choose in Corollary 6.3.4.
We do not know whether this statement continues to hold if the assumption that or is quadratic étale over is dropped. In this case, the fact that is semilocal implies that two Azumaya algebras of the same degree are isomorphic [ojanguren_71]. With this in hand, Corollary 6.3.4 implies that if is equivalent to an Azumaya -algebra admitting a -involution, then has a -involution. The problem therefore reduces to the question of whether the existence of a -involution on implies the existence of a -involution on . The same question was asked for arbitrary non-commutative semilocal rings in [first_15, §12], where it was also shown that counterexamples, if any exist, are restricted. In particular, returning to the case of Azumaya algebras, it follows from [first_15, Thm. 7.3] that if is even, then does posses a -involution when has one.
6.4. The Kernel of the Transfer Map
We continue to use , , , defined in Subsection 6.3.
Saltman’s theorem can also be regarded as a result characterizing the kernel of the transfer map in terms of existence of certain involutions. We now use Theorem 6.3.3 to generalize this particular aspect, namely, describing the kernel of in terms of the involutions that the Brauer classes support. For that purpose, we introduce the following families of -involutions.
Definition 6.4.1.
Let be an Azumaya -algebra. A -involution is called semiordinary if there exists a split Azumaya -algebra and a -involution such that and are locally isomorphic. If can moreover be chosen to be and there is such that is given by on sections, we say that is ordinary.
When is not semiordinary, we shall say it is extraordinary.
Theorem 6.4.2.
With notation as in Theorem 6.3.3, let be an Azumaya -algebra of degree . Then the following conditions are equivalent:
-
(a)
,
-
(b)
there exists admitting a semiordinary -involution,
-
(c)
there exists admitting a ordinary -involution.
In (b), the algebra can be chosen to satisfy and to have a semiordinary involution of any prescribed coarse type in . In (c), the algebra can be chosen to satisfy and to have an ordinary involution of any prescribed coarse type in .
We shall see below (Corollary 6.4.7) that in the situation of a scheme on which is invertible and a trivial involution, or a quadratic étale covering of schemes with its canonical involution, all -involutions are ordinary. Thus, Theorem 6.4.2 recovers Saltman’s Theorem when is invertible.
More generally, it will turn out that under mild assumptions, all involutions are ordinary when is unramified or everywhere ramified.
Proof.
As in the proof of Theorem 6.3.3, we switch to Azumaya -algebras by applying .
(c)(b) is clear.
(b)(a): Suppose admits a semiordinary involution and let be a split Azumaya -algebra with a -involution that is locally isomorphic to . Then by Theorem 6.3.3 and Proposition 5.2.9, .
(a)(c): Let . Then is the image of some . We revisit the proof of the “if” part in Theorem 6.3.3 and apply it with our and to obtain an Azumaya -algebra with involution such that , , and is isomorphic to with being given by on sections. Since , the involution descends to an involution on , defined by the same formula as , hence is ordinary.
It remains to show that we can choose to have a semiordinary involution with a prescribed coarse type . Let be a covering such that lifts to some . Again, we apply the proof of the “if” part of Theorem 6.3.3 with , and to obtain an Azumaya -algebra with involution satisfying and . We then reapply the proof with in place of to obtain another Azumaya -algebra with involution such that is split and . By construction, after suitable refinement of , both and are isomorphic to , where is given by on sections. Consequently, and are locally isomorphic and therefore is semiordinary. ∎
We now shift our attention from the involutions to the coarse types .
Definition 6.4.3.
Let be a coarse -type. We say that is realizable if there exists some Azumaya -algebra and some -involution with coarse type . We also say that is realizable in degree when can be chosen so that . When can be chosen to be ordinary, resp. semiordinary, we call ordinary, resp. semiordinary.
The following theorem characterizes the realizable, semiordinary, and ordinary coarse types in cohomological terms.
Theorem 6.4.4.
With notation as in Theorem 6.3.3, let be coarse type and let , and be as in Subsection 6.3. Then:
-
(i)
is realizable if and only if .
-
(ii)
is semiordinary if and only if .
-
(iii)
is ordinary if and only if , or equivalently .
When (ii) or (iii) hold, is realizable in degree , and hence in all even degrees.
Proof.
Corollary 6.4.5.
With the notation of Theorem 6.3.3, suppose has square roots locally, and assume further that or is unramified. Let be an Azumaya -algebra with a -involution. Then is ordinary, resp. semiordinary, if and only if its coarse type is.
Proof.
The “only if” part is clear, so we turn to the “if” part. We replace with , see Theorem 4.3.11 and Corollary 4.3.14, and write . In case is not connected, we express as such that has degree , and work with each component separately. We may therefore assume that is constant.
By Theorem 5.2.13, it is enough to find an Azumaya -algebra with an ordinary, resp. semiordinary, involution such that and . If is odd, then by Theorem 5.2.17(iii), and we can take . Otherwise, , and applying Theorem 6.4.2 to yields an algebra with an ordinary, resp. semiordinary, involution such that and ; here we used parts (ii) and (iii) of Theorem 6.4.4. ∎
Corollary 6.4.6.
Proof.
Corollary 6.4.7.
With the notation of Theorem 6.3.3, suppose that has square roots locally and moreover
-
(1)
is everywhere ramified and , or
-
(2)
is unramified.
Then all -involutions are ordinary.
Corollary 6.4.8.
Let be a scheme, let be an involution, and let be a good quotient relative to . Assume is noetherian and regular, and is quadratic étale on the generic points of . Then all coarse -types are realizable and semiordinary. If moreover is invertible on , then all -involutions are semiordinary.
Proof.
We conclude this section with two problems, both of which are open both in the context of varieties over fields of characteristic different from with (ramified) involutions and in the context of topological spaces with (non-free) -actions.
Problem 6.4.9.
Is there an element that is not the coarse type of any Azumaya algebra with -involution?
Problem 6.4.10.
Is there an Azumaya algebra with a -involution that is extraordinary (i.e. not semiordinary)?
By Theorem 5.4.5, the first problem can be phrased as follows: Suppse that is a scheme with involution admitting a good quotient relative to , or is a Hausdorff topological space with involution . Let be the locus of points where ramifies and let be a partition of into two closed subsets. Is it always possible to find an Azumaya algebra over admitting a -involution such that the specialization of to is orthogonal if and symplectic if ?
7. Examples and Applications
Example 7.1.1.
Fix an exact quotient and write , . We assume that has square roots locally and . This assumption allows us to drop the distinction between types and coarse types for the most part (Corollary 5.2.14). As in the previous section, we use the notation for the kernel the -norm map , and for the quotient of by the image of the map given by . The coarse types are then .
Suppose is an ordinary coarse type. By Theorem 6.4.4, this is equivalent to saying there exists some in mapping to under the map . Such an can always be found if vanishes, for instance.
Let be a natural number. Consider the matrix
It is immediate that . This equality implies that the map given on sections by
is a -involution. The (coarse) type of is easily seen to be , the image of in . This follows from Construction 5.2.5.
Example 7.1.2.
As a special case of the previous example, we describe the Azumaya algebras with symplectic involution on a scheme or topological space with trivial involution. See Theorem 4.4.4 for the specific hypotheses on the underlying geometric object, and note that we assume is invertible.
In this case, , , and . By Theorem 5.2.17(i), the group of coarse types is , which is just when is connected. We consider the (coarse) type , called the symplectic type.
Any Azumaya algebra with involution having this type must be of even degree, , by Theorem 5.2.17(iii), and is locally isomorphic to the split degree- algebra with symplectic involution
Example 7.1.3.
Fix an exact quotient with ring objects , and let and . Let be a natural number and assume that the hypotheses of Theorem 5.2.13 hold, namely has square roots locally, and either , or is unramified, or is odd. We consider the trivial type, . This is the type of the involution
on the split algebra .
Any algebra with involution of the trivial type is locally isomorphic to this one, and therefore, as summarized in Corollary 5.2.19, these are classified by torsors where . We write the latter group as
and call it the projective unitary group of rank for the involution . In accordance with this notation, the unitary group of will be denoted .
Example 7.1.4.
Consider the case of a scheme or a topological space with trivial involution, as in the case of Example 7.1.2. The theory of Azumaya algebras with involution of type can be established along the same lines as that of type . These algebras are called orthogonal. The automorphism group of is the quotient group , which we denote by , the projective orthogonal group. This is special notation for the group of Example 7.1.3.
Example 7.1.5.
In this example we discuss unitary involutions. As a special case of Example 7.1.3, we consider the case of an unramified double covering of schemes or topological spaces. Again, we refer to Theorem 4.4.4 for the specific hypotheses on the underlying geometric object.
In this case, the ring object is a quadratic étale extension of , see Propositions 4.5.3 and 4.5.4. Since is unramified, Theorem 5.2.17(ii) implies that there is only one type of involution on Azumaya algebras, the trivial one, which is called unitary in this context. In particular, we are in a special case of Example 7.1.3.
The structure of the groups and depends on the nature of , so a complete description in the abstract is not possible. We can, however, find an étale, resp. open, covering such that . After specializing to , the algebra becomes and the involution is becomes the involution given sectionwise by . From this one verifies that and . Consequently, for a general degree- Azumaya -algebra with involution , the groups and are locally isomorphic to and , respectively.
This agrees with the well established fact that projective unitary group schemes of unitary involutions are of type .
Example 7.1.6.
In another instance of Example 7.1.1, we can produce an example of an Azumaya algebra with an involution that mixes the various classical types. This example also featured in the introduction.
We work with étale sheaves and étale cohomology throughout, see Example 4.3.3. Let be an algebraically closed field and let with the -linear involution sending to . The good quotient of by this involution exists, and is given by where .
Here, the ring object is the ring viewed as a sheaf of rings on , and the ring object is the structure sheaf of . The sheaf is the sheaf of norm- elements in , where the norm map sends a Laurent polynomial to . Both the Picard and the Brauer groups of and vanish, so that we can calculate . The following sequence is therefore exact
Explicitly, we calculate that consists of classes of monomials for , and consists of monomials of the form for , but maps the class of to . Therefore, the group of (coarse) types is isomorphic to the Klein -group: .
Since , we are in the circumstance of Example 7.1.1 which provides models for each of the four types on even-degree split algebras. For instance, on , we have the involution given by conjugating by and then applying , namely
Away from the fixed locus of , namely, the points and , this involution is unitary, whereas at it specializes to be orthogonal and at to be symplectic.
More generally, it follows from Theorem 5.4.5 that the type of any Azumaya -algebra with involution is determined by the types seen upon specializing to and .
Example 7.1.7.
We now demonstrate that there exist involutions that are not locally isomorphic to involutions of the form exhibited in Example 7.1.1. Specifically, we will show that there are involutions which are not ordinary in the sense of Definition 6.4.1.
We consider a complex hyperelliptic curve of genus and a double covering . Explicitly: Let be distinct complex numbers, and let
We complete by gluing it to by mapping to , and denote the resulting smooth complete curve by . View as the gluing of to via . Projection onto the or coordinate induces a double covering with ramification at the points and . The map given by , resp. , on the charts is an involution and is a good quotient relative to . Indeed, working with the affine covering , we see that is the fixed ring of
and similarly on the other chart.
By Corollary 6.4.8, all coarse -types in are realizable and semiordinary. Since the branch locus of consists of points, it follows from Theorem 5.4.5 that there are -types. Theorem 6.4.4 also says that the number of ordinary types is the cardinality of the image of the map , where is the sheaf of sections of -norm in . Let . Then . Since is a complete complex curve, the global sections of the structure sheaf are constant functions, meaning that . Since , it follows that . The images of in are therefore the ordinary types. Thus, of the possible coarse types, only are ordinary, and the remaining are merely semiordinary.
We remark that we have reached the latter conclusion without actually constructing Azumaya algebras with involution realizing any of the non-ordinary types. A construction is given in the proof of Theorem 6.4.4, and it can be made explicit in our setting with further work.
This example can also be carried with the affine models and . One can check directly that and thus it is still the case that . Since the ramification point was removed, in this case, there are coarse types, all semiordinary, of which only are ordinary.
Example 7.1.8.
A surprising source of examples comes from Clifford algebras of quadratic forms with simple degeneration. We refer the reader to [auel_fibrations_2014, §1] or [auel_parimala_suresh_2015, §1] for all relevant definitions.
Let be a scheme on which is invertible and let be a line-bundle-valued quadratic space of even rank over ; when and , these data merely amount to specifying a quadratic space of rank over the ring . According to [auel_parimala_suresh_2015, Dfn. 1.9], is said to have simple degeneration if for every , the specialization of to is a quadratic form whose radical has dimension at most . In this case, it shown in [auel_parimala_suresh_2015, Prp. 1.11] that the even Clifford algebra , which is a sheaf of -algebras, is Azumaya over its centre . Furthermore, the sheaf corresponds to a flat double covering , which ramifies at the points where is degenerate. As such, is a good quotient relative to the involution induced by the involution of given by on sections. Abusing the notation, we realize as an Azumaya algebra over .
Suppose has simple degeneration. We moreover assume that is integral, regular and noetherian with generic point and that is nondegenerate, although it is likely that these assumptions are unnecessary. The algebra has a canonical involution , see [auel_mpim_2011, §1.8], and by applying [knus_book_1998-1, Prp. 8.4] to , we see that is of the first kind when and a -involution when . Furthermore, in the case , we have because by [knus_book_1998-1, Thm. 9.12], and is injective by [grothendieck_groupe_1968, Cor. 1.8] (or [auslander_brauer_1960, Thm. 7.2] in the affine case). It therefore follows from Theorem 6.3.3 that there exists with that admits a -involution. We expect that the choice of and its involution can be done canonically in , and with no restrictions on .
With the observations just made, it is possible that our work could facilitate the study of Clifford invariants of non-regular quadratic forms, e.g. in [voight_2011], [auel_fibrations_2014], [auel_parimala_suresh_2015].
8. Topology and Classifying Spaces
The remainder of this paper is concerned with constructing a quadratic étale map of complex varieties and an Azumaya algebra over over such that is Brauer equivalent to an algebra with a -involution, being the non-trivial -automorphism of , but such that the smallest degree of any such is .
We recall that in this particular case, a Brauer equivalent algebra of degree admitting a -involution is guaranteed to exist by a theorem of Knus, Parimala and Srinivas [knus_azumaya_1990, Th. 4.2]; this has been generalized in Theorem 6.3.3. An analogous example in which is the trivial involution was exhibited in [asher_auel_azumaya_2017].
The example, which is constructed in Section 9, will be obtained by means of topological obstruction theory, similarly to the methods of [antieau_unramified_2014], [asher_auel_azumaya_2017] and related works. That is, the desired properties of above will be verified by establishing them for the topological Azumaya algebra over the complexification , whereas the latter will be done by means of certain homotopy invariants.
This section is foundational, describing in part an approach to topological Azumaya algebras with involution via equivariant homotopy theory. The main points are that Azumaya algebras with involution correspond to principal -bundles with involution—a fact that is true even outside the topological context, but that we have not emphasized until now—, that there are equivariant classifying spaces for such bundles, and that their theory is tractable if one restricts to considering spaces on which the -action is trivial or free.
8.1. Preliminaries
In this section and the next, all topological spaces will be tacitly assumed to have a number of desirable properties. All spaces appearing will be assumed to be compactly generated, Hausdorff, paracompact and locally contractible.
Throughout, we work in the category of -topological spaces and -equivariant maps. There are two notions of homotopy one can consider for maps in this setting, the fine, in which homotopies are themselves required to be -equivariant, and the coarse, where non-equivariant homotopies are allowed. These two notions each have model structures appropriate to them, the fine and the coarse. In the fine model structure, the weak equivalences are the equivariant maps inducing weak equivalences on fixed point sets where is either the group or the subgroup . In the coarse structure, it is required only that be a weak equivalence when the -action is disregarded, that is, only the subgroup is considered. The identity functor is a left Quillen functor from the coarse to the fine. This is a synthesis of the theory of [dwyer_singular_1984] with [elmendorf_systems_1983].
Notation 8.1.1.
The notation is used to denote the set of maps between two (possibly unpointed) objects and in a homotopy category. The notation will be used to denote the set of maps between and in the fine -equivariant homotopy category, whereas will be used for the coarse structure.
Remark 8.1.2.
In the case of the coarse model structure, the cofibrant objects include the -CW-complexes with free -action, and if is a -CW-complex, then the construction furnishes a cofibrant replacement of .
All spaces are fibrant in both the coarse and the fine model structures, which implies the following standard result.
Proposition 8.1.3.
If is a free -CW-complex and is a -space, then there is a natural bijection
It is well known that -equivariant homotopy theory in the coarse sense is equivalent to homotopy theory carried out over the base space . We refer to [shulman_parametrized_2008, Sec. 8] for a sophisticated general account of this equivalence. Specifically, the Borel construction and the relative mapping space form a Quillen equivalence between -equivariant spaces with the coarse structure, and spaces over , endowed with what [shulman_parametrized_2008] calls the “mixed” structure on spaces over .
Proposition 8.1.4.
Suppose and are -spaces with being a -CW-complex. Then the Borel construction gives rise to a natural bijection
8.2. Equivariant Bundles and Classifying Spaces
There is a general theory of equivariant bundles and classifying spaces, more general indeed than what is required in this paper. All examples we consider are of the following form:
Definition 8.2.1.
Suppose we are given a topological -group , or equivalently, a topological group equipped with an involutary automorphism . A principal -bundle with a -involution, or just principal -bundle with involution, on a -space is a map in -spaces such that:
-
(1)
is a principal -bundle,
-
(2)
the actions of and of on are compatible, in the sense that if , and , then
Remark 8.2.2.
This concept admits an equivalent definition. Any -bundle , equivariant or not, may be pulled back along the involution of , in order to form . One may then twist the the -action on by changing the structure group along , forming . This may be identified with as a topological space over , but with a different -action. The definition of principal -bundle with involution given above is equivalent to asking that be a principal -bundle together with a -bundles morphism of order from to . On the underlying spaces, must be an isomorphism of order of over , which is equivalent to a -action on making equivariant. The fact that is an isomorphism of principal -bundles is exactly the relation above.
Because the automorphism is not assumed to be trivial, this notion is more general than the most basic notion of ‘equivariant principal -bundle’, but at the same time, because the sequence
is split, it is less general than the most general case considered in [may_remarks_1990].
One may construct a -equivariant classifying space for -equivariant principal -bundles, as in [may_remarks_1990]*Thm. 5. We will take the time to explain the procedure, since some of the details will be important later 111We remark that in our case, the group called in [may_remarks_1990] is a semidirect product, so , with an appropriate -action, is a model for . This allows us to replace the space of sections of by the space of maps , an argument that appears in [Guillou2017]*Sec. 5, p. 21.
Notation 8.2.3.
The notation will be used for a construction of the classifying space of a topological group , functorial in .
By functoriality, if admits a -action, then admits a -action. While the ordinary homotopy type of is well defined, irrespective of the model we choose, the -equivariant type is not. The construction outlined below is a specific choice of such a type.
Start with a . Now consider the space of continuous functions . It is endowed with both a -action, induced directly by the -action on , and by a -action given by conjugation of the map. The two actions together induce an action of on , which is contractible, and consequently a -action on , which is a model for . The resulting map
is a map of -spaces, and will be denoted
We remark that in [may_remarks_1990] and other sources, May and coauthors denote these spaces and .
Furthermore, the map induces a map . This map is -equivariant, and induces a -equivariant commutative square
| (17) |
in which the horizontal maps are coarse, but not necessarily fine, -weak equivalences. The map is a classifying space for principal -bundles with involution.
Proposition 8.2.4.
If is a -CW-complex, then there is a natural bijection between and the set of isomorphism classes of principal -bundles with involution on .
We refer to [may_remarks_1990, Thm. 5] for the proof.
Proposition 8.2.5.
If is a free -CW-complex, then the following are naturally isomorphic
-
(a)
,
-
(b)
,
-
(c)
,
-
(d)
,
-
(e)
The set of isomorphism classes of principal bundles with involution on .
Remark 8.2.6.
Proposition 8.2.5 means that if one is willing to restrict one’s attention to spaces with free -action, then the construction of from is not necessary. The -action given by the functoriality of the construction of is sufficient.
Remark 8.2.7.
Let be a topological group. One may give the -action which interchanges the two factors. Then the resulting classifying space also admits this action. In this instance, the space is -equivalent to with the interchange action, which may be verified by testing on -fixed points, for instance.
The construction of taking a space and producing with the -action interchanging the factors is right adjoint to the forgetful functor. Suppose is a -space, then
| (18) |
where the set on the left is the set of maps in the nonequivariant homotopy category.
8.3. The Case of -bundles
For the rest of this section, we write , etc. for the Lie group of complex points, , and so on.
We now specify -actions on groups that will appear in the sequel. There is a -action on in which the non-trivial element acts via , the transpose-inverse. This passes to certain subquotients of , and we will use it as the -action on the groups , , and , all viewed either as subgroups or as quotients of . Specifically, we write for the outer automorphism .
There is also a -action on given by interchanging the factors and then applying the transpose-inverse, so that the induced involution is
This will be used for certain subquotients of this group, including , , and .
There is a diagonal inclusion , given by . It is -equivariant, and induces similar maps for the aforementioned subquotients of .
One may form -equivariant classifying spaces for the groups named above, as outlined in Subsection 8.2. Among the possibilities, two are particularly useful to us: and .
Proposition 8.3.1.
Let be a -CW-complex with corresponding involution , and let be a natural number. Then the following sets are in natural bijective correspondence:
-
(a)
Isomorphism classes of degree- topological Azumaya algebras with -involution on ,
-
(b)
Isomorphism classes of principal -bundles with involution on ,
-
(c)
.
Proof.
There is a well-known bijection between Azumaya algebras of degree on and principal -bundles, since is the automorphism group of as a -algebra, see Subsection 2.5. Let be an Azumaya algebra of degree on and the associated principal -bundle.
The functor of taking opposite algebras on Azumaya algebras corresponds to the functor of change of group along of principal -bundles; this can be seen at the level of clutching functions. Indeed, note that is a -algebra isomorphism. If one chooses coordinates for on two open sets of on which it trivializes, then the clutching function given by , for some . For the same choice of coordinates over both and , the clutching function of the opposite algebra is given by .
Therefore, the data of an isomorphism of of order over the involution is equivalent to an order- self-map of the associated principal -bundle, over , where denotes the principal -bundle
As explained in Remark 8.2.2, this is equivalent to the definition of principal -bundle with involution in Definition 8.2.1; thus establishing the equivalence of (a) and (b).
The space , by similar methods, is seen to classify ordered pairs of -bundles on a -space , such that the one is obtained from the other by twisting relative to the involutions of and . But the category of such ordered pairs is identical to the category of ordinary -bundles on the space , forgetting the -action.
This last fact also manifests itself algebraically via Remark 8.2.7 in the following way: Suppose is a subgroup of closed under taking transposes, or a quotient of by such a subgroup, let denote the product group with the involution , and let denote the product group with the involution exchanging and . Then is a -equivariant isomorphism between these two groups with involution.
We will apply the classifying space theory developed above in the two extreme cases where the -action on is trivial and when it is free.
8.4. Trivial Action
Suppose is equipped with a trivial -action. Then principal -bundles with involution on are classified by .
Proposition 8.4.1.
Let be a positive integer. Then the fixed locus is homeomorphic to
-
(i)
if is even;
-
(ii)
if is odd.
Proof.
We may calculate the fixed-point-sets of by means of [may_remarks_1990]*Thm. 7. We explain the application of this theorem in the current case.
If is a matrix such that , then generates a subgroup that maps isomorphically onto and intersects trivially. Denote by the commutant of in , i.e., the subgroup of consisting of elements such that . We write if and are conjugate under , or equivalently, if there exists such that . Then the theorem asserts that
as runs over equivalence classes of elements satisfying .
When is even, say , there are two such equivalence classes, namely the class of and the class of , in the notation of Example 7.1.1, as can be calculated directly. The fixed points under the action are those matrices for which in the first case and in the second, which is to say, the subgroups of orthogonal and of symplectic matrices respectively. We therefore deduce
When is odd, the argument is much the same, but only occurs. ∎
Remark 8.4.2.
By Theorem 5.4.5, we know that there are two types of involutions on Azumaya algebras over connected topological spaces with trivial action, the symplectic and orthogonal. By means of Examples 7.1.2 and 7.1.4, we know that the orthogonal and symplectic Azumaya algebras with involution are equivalent to principal bundles for the groups and , the latter when is even. Proposition 8.4.1 has recovered these observations via equivariant homotopy theory.
8.5. Free Action
Now we address the case where the action of on is free. In this case, the quotient map is a two-sheeted covering space map.
Proposition 8.5.1.
Let be a free -CW-complex, with acting by the involution , and let . Consider as a space over , or alternatively, as a space equipped with a distinguished class . There are natural bijections between the following:
-
(a)
Isomorphism classes of degree- topological Azumaya algebras over equipped with a -involution.
-
(b)
.
-
(c)
.
-
(d)
Elements of the preimage of under .
Proof.
We continue to assume that is a free -CW-complex and let . We would like to have a classifying-space-level understanding of the cohomological transfer map considered in Subsection 6.2.
To that end, let denote the discrete group or the topological group . We endow with the involution , give the involution , and let denote with the trivial action.
The map defined by is -equivariant, and its kernel consists of pairs of the form , which is the image of the diagonal map . That is, there is a -equivariant short exact sequence of -groups
and therefore, a sequence of -equivariant maps in which any three consecutive terms form a homotopy fibre sequence:
Any such homotopy fibre sequence is a homotopy fibre sequence in the -equivariant coarse structure. These constructions are plainly natural with respect to inclusion of subgroups of .
Now, if is a free -CW-complex, then thanks to Proposition 8.2.5, one arrives at a long exact sequence of abelian groups
| (19) |
Since with this action is isomorphic to with the interchange action, it follows that . Moreover, is simply .
Therefore, the sequence of (19) reduces in this case to
| (20) |
When and , the map denoted agrees with the transfer map defined in Subsection 6.2. Indeed, we know that the transfer map in Section 6 agrees with the ordinary transfer map for a -sheeted covering in the case at hand, Example 6.2.5. It suffices therefore to show that the map in (20) is the usual transfer map for a -sheeted cover. The trivial case is elementary. The general case where is merely locally trivial can be deduced from the trivial case by viewing and as Čech cohomology groups and calculating each using covers of and of where trivializes the double cover .
Proposition 8.5.2.
Let be or , given the involution . Let be a space with free -action, let , and let be an equivariant map, representing a cohomology class . Then .
Proof.
Since is equivariant, and the action on is free, lies in the image of in . Thus, the result follows from the exact sequence (20). ∎
9. An Azumaya Algebra with no Involution of the Second Kind
We finally construct the example promised at the beginning of Section 8.
Throughout, the notation means a free cyclic group, written additively, with a named generator . Recall that for a topological space , the sheaf cohomology group is isomorphic to the singular cohomology group , see [asher_auel_azumaya_2017, §2.1], for instance. We shall use the latter group for the most part.
9.1. A Cohomological Obstruction
In all cases, the groups appearing in this subsection are the complex points of linear algebraic groups. In the interest of brevity, the relevant linear algebraic group, e.g. , will be written in place of the group itself, e.g. .
Unless otherwise specified, groups appearing will be endowed with a -action. For the groups , the action is that sending to , which restricts to the action on the central subgroup . For the groups , the action is that given by , and similarly for . The maps and are given by diagonal inclusions.
Embed via , and let denote the group obtained as the quotient of by the image of .
In the following diagram, the horizontal arrows of the first two rows are -equivariant. This induces -actions on the groups in the third row so that all arrows become -equivariant.
Each of the groups appearing above is equipped with a -action, and consequently each may be extended to a semidirect product with , and equivariant classifying spaces of the form may be constructed as in Subsection 8.2. Since we will consider equivariant maps with free -action on the source, by Proposition 8.2.5, we may use any functorial model of with its functorially-induced -action instead.
Proposition 9.1.1.
The -action on induces an action on . In low degrees, this action is summarized by Table 1.
| Action | ||
| 0 | trivial | |
| 1 | 0 | - |
| 2 | 0 | - |
| 3 | ||
| 4 | trivial |
Proof.
The compatible -actions on the terms of the exact sequence induce an action on the fibre sequence , and therefore an action of on the associated Serre spectral sequence, which is illustrated in Figure 2.
Here, is if is even and is otherwise.
The action on is the same as the action of on itself, which is the sign action. The action on is calculated by identifying as a subquotient of , where is the maximal torus of diagonal matrices in . Specifically, , where the -action on is . Then the class in question may be identified with the image of the second elementary symmetric function in the in . It follows the action of on is trivial.
We know from [antieau_topological_2014, Proposition 4.4] that the illustrated differential is surjective. Writing for , it follows easily that the cohomology of takes the stated form, and carries the stated -action. ∎
Proposition 9.1.2.
Fix a natural number , and let . Let be the subgroup of consisting of terms where . The low-degree cohomology of , along with its -action, is summarized by Table 2.
| Action | ||
| 0 | trivial | |
| 1 | 0 | - |
| 2 | 0 | - |
| 3 | ||
| 4 | S |
Moreover, the comparison map from to is the evident identification map when . When , it is given by .
Proof.
There is a fibre sequence .
A portion of the associated Serre spectral sequence is shown in Figure 3. There is a comparison map of spectral sequences from this one to that of Figure 2. The map identifies the bottom row of the two -pages, and sends both to . It is compatible with the -actions. The claimed results except the -action on all follow from the comparison map and the values in Table 1. As for the action on , as in the proof of Proposition 9.1.1, this can be deduced from the action on , which is given by and . ∎
Remark 9.1.3.
From Figures 2 and 3, we deduce that the maps and both represent generators of the groups and , respectively. Moreover, the image of the former class under the Bockstein map is a generator of , which is nothing but the tautological Brauer class of . That is, if is the classifying map for a -bundle, or equivalently, a degree- topological Azumaya algebra, then the Brauer class of that algebra is .
Our purpose in introducing the group is to construct a group which is as close to (with the interchange action) as possible, but for which the transfer of all classes in vanish.
Proposition 9.1.4.
Let be as constructed above. Give the space the diagonal -action. Then the transfer map, , considered at the end of Subsection 8.5, vanishes.
The space is weakly equivalent to , but carries a free -action.
Proof.
Proposition 9.1.5.
Let be an even integer, and let be an odd integer. Suppose is a -equivariant map and a -equivalence. Then there is no -equivariant map inducing a surjection on .
Proof.
For the sake of contradiction, suppose that exists.
By Remark 9.1.3, the comoposition
induced by and the inclusion , represents a generator of . As a result, there is such that the composition
represents . Consequently, the map fits into a homotopy-commutative square
in which is the composition of and the map induced by . We extend this square into a homotopy commutative diagram
where both rows are homotopy fibre sequences, so is the homotopy fibre of . Strictly speaking, we carry this out in the (fine) -equivariant model structure on topological spaces, using the dual of [hovey_model_1999, Prop. 6.3.5] to deduce the existence of the dashed arrow in that category, so that it may be assumed to be -equivariant. Moreover, the space appearing in this argument has the appropriate non-equivariant homotopy type, since the functor forgetting the -action is a right Quillen functor, and therefore preserves fibre sequences.
Each of the two fibre sequences is associated to a Serre spectral sequence in cohomology. In the case of the lower row, the -page is represented in Figure 2, whereas in the case of the upper row, since is -equivalent to , it is isomorphic on the -page to the spectral sequence represented in Figure 3. There is an induced map between these spectral sequences, and this map restricts to the following on the -line:
We know the map on is surjective, because in each case, the group is generated by a class for which is , obtained by taking the canonical class in , resp. , squaring it, and applying the Bockstein map with image . This may be deduced from [cartan_determination_1954], or from the Serre spectral sequence associated to the path-loop fibration .
Since the map of spectral sequences is compatible with the -action, it induces the following commutative square
in which all arrows are -equivariant. Furthermore, the proofs of Propositions 9.1.1 and 9.1.2 imply that both horizontal maps are surjective, that acts trivially on , and , and that the non-trivial element of interchanges and . Now, lies in the -fixed subgroup of , which is to say for some integer . Then is times a generator of , and hence not a generator of . On the other hand, is a generator of by the previous paragraph, a contradiction. ∎
9.2. An Algebraic Counterexample
In this section, we consider complex algebraic varieties. In particular, all algebraic groups are complex algebraic groups. Cohomology is understood to be étale cohomology in the context of varieties and singular cohomology in the context of topological spaces.
A -action on a variety will be called free if there exists a -torsor . In this case, coincides with categorical quotient in the category of varieties. Furthermore, if acts freely on , then it also acts freely on . The converse holds when is affine or projective, see Example 4.3.3 and Proposition 4.5.3, but not in general.
Fix an even positive integer . We define the complex algebraic group by means of the short exact sequence
so that is the group considered in the previous subsection. There is a natural map . Composing with the map induced by the inclusion allows us to associate with every -torsor an -torsion class in . This association is natural, and is, in particular, compatible with complex realization.
The first projection induces a group homomorphism (it is not -equivariant). Using this map, we associate to every -torsor a -torsor, namely, .
Lemma 9.2.1.
With the previous notation, let be a -torsor, let be its associated class in , and let be its associated -torsor. Then is the image of under the canonical map . In particular, .
Proof.
This follows by considering the following morphism of short exact sequences and the induced morphism between the associated cohomology exact sequences.
Note that the vertical maps are not necessarily -equivariant. ∎
Proposition 9.2.2.
Maintaining the previous notation, there exists a smooth affine complex variety with free -action, a -torsor and a map such that the following hold:
-
(i)
The map is -equivariant and a -equivalence.
-
(ii)
The homotopy class of corresponds to the principal -bundle .
-
(iii)
The Brauer class associated with has trivial image under .
For later reference, and in keeping with the previous parts of this paper, we denote by .
Proof.
As in Subsection 9.1, the group is an affine algebraic group equipped with an algebraic -action. Consequently, the split exact extension in
is also an affine algebraic group, [molnar_semi-direct_1977]*Ex. 2.15 (c).
Therefore it is possible to follow [totaro_chow_1999] and construct affine spaces on which acts and such that becomes a -torsor after removing a locus, , of arbitrarily large codimension. In particular, is a -torsor. Choose so that has (complex) codimension at least .
Let and , and note that both and carry -actions. The -action on is free since is a -torsor. Moreover, one checks directly that is a principal -bundle with involution, see Definition 8.2.1. Since acts freely on , Proposition 8.2.5 implies that this principal bundle is represented by a map , which satisfies conditions (i) and (ii).
By means of the equivariant Jouanolou device, [hoyois_six_2017]*Prop. 2.20, we may assume that is a smooth affine variety with these properties.
Let denote the Brauer class associated with . It remains to show that in , where .
To that end, let be the image of under , and similarly define as the image of under the analogous map in singular cohomology. It is enough to check that vanishes. There is a commutative diagram
where each map from left to right is a transfer map, each map from back to front is a complex-realization map, and the maps from top to bottom are induced by the inclusion . The two indicated maps are isomorphism by Artin’s theorem. We also remark that , where the first group is understood as sheaf cohomology with coefficients in the sheaf of nonvanishing continuous complex-valued functions. Now, the transfer of is easily seen to be by comparison with , where it is known to vanish by Proposition 9.1.4. This completes the proof. ∎
Theorem 9.2.3.
For any even integer , there exists a quadratic étale map of smooth affine complex varieties and an Azumaya algebra of degree over such that:
-
(i)
The period and index of are both .
-
(ii)
in .
-
(iii)
The degree of any Azumaya algebra Brauer equivalent to and admitting a -involution is divisible by —here denotes the non-trivial involution of over .
In particular, we see that the minimal degree of an Azumaya algebra Brauer equivalent to and supporting a -involution is at least . This bound is sharp by Theorem 6.3.3.
Proof.
Construct as in Proposition 9.2.2, and let be the Azumaya algebra corresponding to the -torsor associated to by means of . The Azumaya algebra has degree , and the complex reaization of its Brauer class is a generator of , since the map induces an isomorphism on , see Remark 9.1.3 and the proof of Proposition 9.1.2. In particular, the period and index of must both be .
Let be an odd integer and suppose were equivalent to an Azumaya algebra of degree carrying a -involution. Then, by Proposition 8.3.1, the complex realization of this algebra would correspond to a topological -bundle with involution. By Proposition 8.2.5, there would be a -equivariant map such that was the image of the canonical Brauer class in , and therefore would induce a surjection in , since the image of would contain . This is forbidden by Proposition 9.1.5. ∎
Question 9.2.4.
Does Theorem 9.2.3 hold when is odd?
Appendix A The Stalks of The Ring of Continuous Complex Functions
Let be a topological space; we work throughout on the small site of . Let denote the sheaf of continuous -valued functions on .
Let be a point of and consider . It is a local ring with maximal ideal denoted . An element is the germ of a continuous -valued function at , and the class is the complex number .
Proposition A.0.1.
The local ring is strictly henselian.
Proof.
It suffices to prove the ring is a henselian ring as the residue field is .
Consider as an ordered set of roots of a degree- monic polynomial. There is a permutation action of the symmetric group on , and there is a homeomorphism , where the map takes to the coefficients of the polynomial , [bhatia_space_1983].
We embed as the permutations fixing the first element, and form .
The space represents monic polynomials of degree and a distinguished ‘first’ root, . There is a closed subset , the locus where . Then is an open subset representing the set of monic, degree- polynomials having a distinguished ‘first’ root which is not repeated. Since quotient maps of spaces given by finite group actions are open maps, in the following diagram, every map appearing is an open map:
We denote the composite map by . It sends a pair , for which , to .
Let be such a pair. We claim that there exists an open neighbourhood of such that is a homeomorphism onto its image. Choose an open neighbourhood of of the form , being the product of an -ball around and around , where is sufficiently small that none of the polynomials in has any of the complex numbers in as roots. It is immediate that is injective when restricted to this open set in . Since is an open map, is a homeomorphism onto its image, establishing the claim.
Suppose we are given a polynomial . Suppose further that a non-repeated root, , of is given, where is the reduction of to . We can write in . Note that we do not yet assert that and are the reductions of any specific elements in or . To prove that the ring is Henselian, we must find an element lifting and satisfying .
The germ has an extension to an open neighbourhood .
We have the data of a diagram
Around the image of in we can find an open set such that is a homeomorphism onto the image, . Then is an open set in containing . Since , the preimage is an open subset of containing . Since is a homeomorphism, we may lift the map
as indicated.
That is to say, there is a neighbourhood, , of such that the factorization can be extended on to a factorization . In particular, the class of in is a root of the polynomial extending . This proves Hensel’s lemma for . ∎